The complex interaction of a stationary background flow and an acoustic field can be modeled using the linearized Navier-Stokes physics interfaces in the Acoustics Module. The interfaces allow for a detailed analysis of how a flow, which can be both turbulent and nonisothermal, influences the acoustic field in different systems. This includes all linear effects in which a background flow interacts with and modifies an acoustic field. The linearized Navier-Stokes interfaces do not induce flow-induced noise source terms. Basically, the equations solve for the full linear perturbation to the general equations of CFD — mass, momentum, and energy conservation.
Being able to model and simulate the details of how a background flow influences an acoustic field is important in many industries and areas of application. In the automotive industry, the acoustic properties of exhaust and intake systems are altered when a flow passes through them. For example, the transmission loss of a muffler changes depending on the magnitude of the bypass background flow. In aerospace applications, the study of how liners and perforates behave acoustically when a flow is present is of high interest. The detailed acoustic properties (absorption, impedance, and reflection coefficients) of these subsystems influence the system-level behavior of, for example, a jet engine.
In the muffler and liner examples, the attenuation of the acoustic signal by the turbulence present in the background flow can also be captured with the linearized Navier-Stokes equations. Moreover, the background flows in these models are often nonisothermal in nature.
Example of an automotive application. Results from the Helmholtz resonator with flow example presented below. In front, the color surface plot is of the sound pressure level. At the back, the streamlines are of the background flow.
The linearized Navier-Stokes interfaces have a built-in multiphysics coupling to structures. This enables an out-of-the-box setup of fluid-structure interaction (FSI) models in the frequency domain (or in the linear regime in the time domain). The interaction of flow, acoustics, and structural vibrations is important in many applications. One application example is for flow sensing in a Coriolis flow meter. In general, these interfaces are suited for the analysis of the changed vibrational behavior of structures when under a fluid load by a background flow.
Example of FSI in the frequency domain: the movement of a Coriolis flow meter actuated at the fundamental frequency. The surface shows the structural deformation (the phase and amplitude are highly exaggerated for visualization) and the open cut-out section of the pipe shows the acoustic pressure on the pipe’s inner surface.
Other applications of the linearized Navier-Stokes interfaces include the study of combustion instabilities and general in-duct acoustics as well as more academic applications like analyzing the onset of flow instabilities or studying regions prone to whistling.
The interfaces now include the Galerkin least squares (GLS) stabilization scheme, enabling more robust simulations. This new default setting better handles the numerical and physical instabilities introduced by the convective and reactive terms included in the governing equations. Moreover, the reformulated slip boundary condition is now well suited when solving models with an iterative solver. This is crucial in cases where large industrial problems have to be solved.
The linearized Navier-Stokes equations represent a linearization to the full set of governing equations for a compressible, viscous, and nonisothermal flow (the Navier-Stokes equations). It is performed as a first-order perturbation around the steady-state background flow defined by its pressure, velocity, temperature, and density (p_{0}, u_{0}, T_{0}, and ρ_{0}). This results in the governing equations for the propagation of small perturbations in the pressure, velocity, and temperature (p, u, and T) — the dependent variables. In perturbation theory, a subscript 1 is sometimes used to express that these variables are first-order perturbations. The governing equations (with subscript 0 on the background fields) read:
(1)
where Φ = ∇u : τ_{0} + u_{0} : τ is the viscous dissipation function; M, F, and Q represent possible source terms; κ is the coefficient of thermal conduction (SI unit: W/m/K); α_{p} is the (isobaric) coefficient of thermal expansion (SI unit: 1/K); β_{T} is the isothermal compressibility (SI unit: 1/Pa); and _{p} is the specific heat capacity (heat capacity per unit mass) at constant pressure (SI unit: J/kg/K).
In the frequency domain, the time derivatives are, in the usual manner, replaced by a multiplication with iω. The constitutive equations for the stress tensor and the linearized equation of state (density perturbation) are given by:
(2)
where τ is the viscous stress tensor (Stokes expression), μ is the dynamic viscosity (SI unit: Pa s), and μ_{B} is the bulk viscosity (SI unit: Pa s).
The Fourier heat conduction law is used in the energy equation. A detailed derivation of the equations can be found in the Acoustics Module User’s Guide. The equations can be solved in the time domain or frequency domain using either the Linearized Navier-Stokes, Transient interface or the Linearized Navier-Stokes, Frequency Domain interface.
By taking a closer look at the governing equations presented in (1), you can see that they contain different types of terms:
Because of the general nature of the equations solved in the interfaces, they naturally model the propagation of acoustic (compressible) waves, vorticity waves, and entropy waves. The latter two types of waves are only convected with the background flow velocity and do not propagate at the speed of sound. As an acoustic wave propagates, it can interact with the flow (through the reactive terms) and energy can be transferred to and from an acoustic mode to both the vorticity and entropy modes. The reactive terms in the governing equations are responsible for this flow-acoustic-like coupling. This is in the sense that the vorticity and entropy waves are nonacoustic (CFD-like) perturbations to the background flow solution, so to some extent, they model the linear interaction between CFD and acoustics.
In many aeroacoustics formulations, the reactive terms are disregarded, as they are also responsible for the processes that generate the Kelvin-Helmholtz instabilities. These can be difficult to handle numerically. On the other hand, if the terms are disregarded, accurate modeling of sound attenuation and amplification is lost. The reactive terms are fully included in the linearized Navier-Stokes interfaces.
The growth of the instabilities is handled in two ways in COMSOL Multiphysics. The temporal growth of the instabilities can be handled by selecting the frequency-domain formulation rather than the time-domain formulation. The spatial instabilities, which can arise if the vorticity modes are not properly resolved, are efficiently handled by the GLS stabilization scheme.
Depending on the application modeled with the linearized Navier-Stokes equations, it may be necessary to resolve the acoustic, viscous, and thermal boundary layers. These are naturally created on solid surfaces for an oscillating flow, when no-slip and isothermal boundary conditions are present. Typically, it is not necessary to include the details of the losses in the boundary layers in large models (when compared with the boundary layer thickness). The thermal boundary layer can also often be disregarded in liquids but should be included with equal importance in gasses. The two effects can be disregarded by selecting either the slip or the adiabatic options on the wall boundary conditions.
It should be mentioned that one more indirect coupling between the background flow and the acoustics is possible. When an acoustic wave propagates through a region with turbulent background flow, it is attenuated. This effect can be included in the model by coupling the turbulent viscosity from the CFD RANS model to the acoustics model. This effect is important, for example, when analyzing the transmission loss of a muffler system in the presence of a flow.
Solving the linearized Navier-Stokes equations, which falls under the field of computational aeroacoustics (CAA), poses numerical challenges that need to be considered, understood, and handled carefully. As mentioned above, the governing equations are prone to both physical (Kelvin-Helmholtz) and numerical instabilities. Because the interfaces use stabilization, the remaining main numerical challenge is to avoid the introduction of numerical noise in terms involving the background field variables (p_{0}, u_{0}, T_{0}, and ρ_{0}). This is especially true in the reactive terms involving the gradient of the background flow variables.
The likelihood of this problem increases if different meshes are used for the CFD and acoustic models and/or different discretization orders are used for the background flow and acoustics problem. Note that using different meshes or discretization orders is well motivated by the fact that the two problems need to resolve different physics and length scales. To prevent this, a careful mapping of the background flow data from CFD to acoustics is necessary. This is a well-understood and described step in CAA modeling. Additionally, the mapping step can be used to smooth the CFD data. This can be an overall smoothing or a local smoothing of certain details, like the hydrodynamic boundary layer, if its details are not important for the acoustics model.
In COMSOL Multiphysics, the mapping between the mesh is performed by an additional study step. The details of this step are described in the Acoustics Module User’s Guide and in tutorial models using a linearized Navier-Stokes physics interface.
When performing simulations with a linearized Navier-Stokes physics interface, the following points should be considered:
Helmholtz resonators (used in exhaust systems) attenuate a narrow and specific frequency band. When a flow is present in the system, it changes the resonator’s acoustic properties as well as the subsystem’s transmission loss. The Helmholtz resonator tutorial model investigates the transmission loss in the main duct (the resonator is located as a side branch) when a mean flow is present.
To calculate the mean flow, the SST turbulence model is used for Mach numbers Ma = 0.05 and Ma = 0.1. The Linearized Navier-Stokes, Frequency Domain interface is used to solve the acoustics problem. Next, the acoustics model is coupled to the mean flow velocity, pressure, as well as turbulent viscosity. The predicted transmission loss shows good agreement with results from a published journal paper (Ref. 1). For the resonances to be located correctly and the amplitude of the transmission loss to be correct, the model must balance convective and diffusive terms properly. This is achieved in the model.
Transmission loss through the resonator as a function of frequency and Mach number of the background flow.
The pressure distribution inside the system at 100 Hz for Ma = 0.1. A plane wave is incident from the left side upstream of the flow.
In the Acoustic Liner with a Grazing Background Flow tutorial model, the acoustic liner consists of eight resonators with thin slits and the background grazing flow is at Mach number 0.3. The sound pressure level above the liner is calculated and shows good agreement with results from a published research paper (Ref. 2). This example computes the flow via the SST turbulence model in the CFD Module and the acoustic propagation with the Linearized Navier-Stokes, Frequency Domain interface. The acoustic boundary layer is resolved and the default linear discretization is switched to quadratic to improve the spatial resolution near walls.
The curves show the sound pressure level on the surface above the liners for four driving frequencies. The colored part of the curves highlights the extent of the liner. These results show good agreement with the experimental results from the referenced research paper.
The acoustic velocity fluctuations as a plane wave propagates above the liners, showing the first four liners. The driving frequency is 1000 Hz. The color plot shows the velocity amplitude and the arrows show the velocity vector. Near the holes at the surface of the liner, vorticity is generated by the flow-acoustics interaction.
Coriolis flow meters — also called mass or inertial flow meters — can measure the mass flow rate of a fluid moving through it. This device can also compute the density of the fluid, hence the volumetric flow rate. The Coriolis Flow Meter tutorial model demonstrates how to model a generic Coriolis flow meter with a curved geometry.
As a fluid travels through an elastic structure (a curved duct, for instance), it interacts with the movement of the structure when vibrating. The Coriolis effect causes a phase difference between the deformation of two points on the duct, which can be used to determine the mass flow rate.
To model this, the Linearized Navier-Stokes, Frequency Domain interface is coupled to the Solid Mechanics interface via the built-in multiphysics coupling. As for the background mean flow, it is simulated with the Turbulent Flow, SST interface. Using this approach, FSI can be efficiently modeled in the frequency domain.
The phase difference between upstream and downstream points (red dots on the animation below). This curve represents the calibration results needed to run a Coriolis flow meter.
The movement of the Coriolis flow meter for three mass flow rates. The flow meter is actuated at the natural frequency of the structure, f_{d} = 163.5 Hz. The deformation amplitude and phase are exaggerated for visualization. As the flow rate increases, the phase difference upstream and downstream increases.
E. Selamet, A. Selamet, A. Iqbal, and H. Kim, “Effect of Flow in Helmholtz Resonator Acoustics: A Three-Dimensional Computational Study vs. Experiments”, SAE International Journal, 2011.
C. K. W. Tam, N. N. Pastouchenko, M. G. Jones, and W. R. Watson, “Experimental validation of numerical simulations for an acoustic liner in grazing flow: Self-noise and added drag”, Journal of Sound and Vibration, p. 333, 2014.
What was life like during the Ediacaran period, around 635 to 541 million years ago? In general, scientists have assumed that the sea-dwelling creatures of this time were almost all stationary, permanently attached to the seafloor. This assumption is sometimes applied to the Ediacaran taxon Parvancorina, a “poster child” for the more bizarre Ediacara biota that lived around 555 million years ago.
If Parvancorina were alive today, you’d be able to balance its entire shield-shaped body (which was about the size of a penny) on the tip of your finger. Looking closer, you would see a distinctive set of ridges on its back that form an anchor- or T-like shape.
An illustrated interpretation of Parvancorina.
Although there are a number of Parvancorina fossils preserved in rocks from Russia and South Australia, this simple organism left no clues about whether or not it moved or how it fed. Solving these mysteries is a key step toward determining the evolutionary and ecological importance of Parvancorina.
Parvancorina minchami (left). Image by Matteo De Stefano/MUSE – Science Museum. Licensed under CC BY-SA 3.0, via Wikimedia Commons. A Parvancorina fossil (right). Image is in the Public domain.
To find answers, a team from Vanderbilt University, Oxford University Museum of Natural History, Natural History Museum of Los Angeles County, and University of Toronto Mississauga turned to CFD simulation.
With the COMSOL Multiphysics® software, these researchers were able to gain insight into the life of the ancient Parvancorina, which is “one of the weirdest and most poorly understood Ediacaran [...] organisms,” according to Dr. Simon A.F. Darroch, one member of the research team. Elaborating on the study, he said: “Fluid dynamics simulations provide the only sensible way to test hypotheses surrounding Ediacaran feeding and movement.”
The researchers created 3D CAD models of Parvancorina based on observations of fossil specimens, which they were able to easily import into the COMSOL® software. These models include a null model of the shield-shaped base of the Parvancorina body as well as full models of three different morphotypes:
An example of the team’s Parvancorina model geometry and CFD simulations. Images copyright © Dr. Imran A. Rahman.
Using COMSOL Multiphysics, the team was able to simulate the typical currents of the shallow marine environment Parvancorina called home and see how its structure influenced its interactions with this environment. In their simulations, the researchers tested various inlet velocities, model orientations, and mesh sizes.
The majority of Ediacaran organisms, like Parvancorina, seemed to prefer living in shallow environments, not the deeper ocean environment like that pictured above.
Dr. Imran A. Rahman, another member of the research team, noted that COMSOL Multiphysics is “a very user-friendly program that is capable of simulating fluid flows around the very complex 3D shapes (i.e., fossils) we are interested in.” The software also enabled them to generate high-quality flow velocity and streamline plots to easily visualize their results.
To determine if Parvancorina was mobile or stationary, the researchers used COMSOL Multiphysics to calculate the drag forces experienced by the Parvancorina models. The results show that the Parvancorina morphotypes experienced different amounts of drag at different orientations.
Since drag can be harmful to organisms living on the seafloor, potentially dislodging or injuring them, the ability to maintain a position that minimized drag, which would require reorienting itself in relation to the shifting currents of its shallow-water environment, would have greatly benefited Parvancorina. As such, the results of the CFD simulations provide good indirect evidence that Parvancorina was mobile during its life. Having a body shape that reduces drag in one direction to the current is a common characteristic of mobile organisms living in environments with variable currents.
Being able to maintain a specific position relative to the current direction could have benefited Parvancorina in other ways. The team’s research shows that flow was not distributed evenly over the surface of Parvancorina for any orientation, morphotype, or current speed. Instead, the external morphology caused recirculated flow to be directed toward localized areas, with the particular area changing depending on the orientation of Parvancorina to the current.
Recirculation zones on the Parvancorina body.
This could be a sign that Parvancorina was a suspension feeder, able to direct the flow of organic material suspended in water toward its feeding structures. If so, Parvancorina would have had to maintain a certain alignment in relation to the current to effectively channel food toward the correct areas of its body. Thus, as with the computed drag forces, this result also indicates that Parvancorina would have benefited greatly from the ability to reorient itself on the seafloor.
2D and 3D flow field plots showing that different Parvancorina orientations create recirculation zones at different positions. When Parvancorina faces the flow, this causes the formation of two recirculation zones just behind the “arms” of the anchor-shaped part of its body (top row). When the oncoming flow hits perpendicular to the central “shank”, the main recirculation takes place behind the shank of the anchor-shaped part of the body of Parvancorina (middle row). Finally, if the “arms” are positioned downstream, the recirculation takes place behind the body of Parvancorina (bottom row). 2D images copyright © Dr. Imran A. Rahman.
The distinctive anchor-shaped ridge of Parvancorina yields another clue that tells us this creature was a suspension feeder. While the ridge does not reduce drag and was likely not a defensive structure, the researchers’ simulations indicated that it was needed for recirculation — the null model of Parvancorina without the ridge did not direct recirculated flow over the body. This matches what we know about extant marine invertebrates. Organisms that have specialized structures with the concave sides facing upstream are often passive suspension feeders. This suggests that such structures may provide a common benefit to suspension feeders, like affecting fluid flow and the ability to collect food.
The team’s previous research suggested that stationary Ediacaran suspension feeders required fluid flow that was directed to the food-capturing areas of the body at any orientation (as they couldn’t reorient themselves). Since the flow of organic material is only directed to the presumed food-capturing areas of Parvancorina at certain orientations to the current, it would have to be mobile to feed efficiently as a suspension feeder. It is also possible that Parvancorina fed in a different way, for instance, by consuming dead organic matter in the sediment on the seafloor. The key point is that, if Parvancorina fed in either of these ways, it would need to have been mobile.
Based on their extensive CFD simulations, the research team was able to provide indirect evidence that Parvancorina was mobile and perhaps had musculature or appendages that have not yet been seen in any fossils. If this is correct, it could mean that Parvancorina is the oldest known creature capable of rheotaxis, the behavior of orienting to face oncoming currents.
These findings agree with an independent study performed by another research group who found that Parvancorina fossils were often aligned to face the current and determined that this represents a rheotactic response.
This work opens the door to the possibility that other Ediacaran organisms may have been mobile — offering a new view of Ediacaran benthic ecosystems. The team plans to continue their investigations by using COMSOL software to research other Ediacaran organisms in future projects. “CFD studies have cast extraordinary new light on the biology and ecology of the Ediacaran biota, which represents the first (and most mysterious) radiation of complex life,” Darroch notes. “Understanding where complex life came from, and how it came about, hinges on us understanding where the Ediacarans fit.”
This post was originally published in 2013. It has since been updated to include all of the turbulence models currently available with the CFD Module as of version 5.3 of the COMSOL® software.
Let’s start by considering the fluid flow over a flat plate, as shown in the figure below. The uniform velocity profile hits the leading edge of the flat plate, and a laminar boundary layer begins to develop. The flow in this region is very predictable. After some distance, small chaotic oscillations begin to develop in the boundary layer and the flow begins to transition to turbulence, eventually becoming fully turbulent.
The transition between these three regions can be defined in terms of the Reynolds number, , where is the fluid density; is the velocity; is the characteristic length (in this case, the distance from the leading edge); and is the fluid’s dynamic viscosity. We will assume that the fluid is Newtonian, meaning that the viscous stress is directly proportional, with the dynamic viscosity as the constant of proportionality, to the shear rate. This is true, or very nearly so, for a wide range of fluids of engineering importance, such as air or water. Density can vary with respect to pressure, although it is here assumed that the fluid is only weakly compressible, meaning that the Mach number is less than about 0.3. The weakly compressible flow option for the fluid flow interfaces in COMSOL Multiphysics neglects the influence of pressure waves on the flow and pressure fields.
In the laminar regime, the fluid flow can be completely predicted by solving Navier-Stokes equations, which gives the velocity and the pressure fields. Let us first assume that the velocity field does not vary with time. An example of this is outlined in The Blasius Boundary Layer tutorial model. As the flow begins to transition to turbulence, oscillations appear in the flow, despite the fact that the inlet flow rate does not vary with time. It is then no longer possible to assume that the flow is invariant with time. In this case, it is necessary to solve the time-dependent Navier-Stokes equations, and the mesh used must be fine enough to resolve the size of the smallest eddies in the flow. Such a situation is demonstrated in the Flow Past a Cylinder tutorial model. Note that the flow is unsteady, but still laminar in this model. Steady-state and time-dependent laminar flow problems do not require any modules and can be solved with COMSOL Multiphysics alone.
As the flow rate — and thus also the Reynolds number — increases, the flow field exhibits small eddies and the spatial and temporal scales of the oscillations become so small that it is computationally unfeasible to resolve them using the Navier-Stokes equations, at least for most practical cases. In this flow regime, we can use a Reynolds-averaged Navier-Stokes (RANS) formulation, which is based on the observation that the flow field (u) over time contains small, local oscillations (u’) and can be treated in a time-averaged sense (U). For one- and two-equation models, additional transport equations are introduced for turbulence variables, such as the turbulence kinetic energy (k in k-ε and k-ω).
In algebraic models, algebraic equations that depend on the velocity field — and, in some cases, on the distance from the walls — are introduced in order to describe the turbulence intensity. From the estimates for the turbulence variables, an eddy viscosity that adds to the molecular viscosity of the fluid is calculated. The momentum that would be transferred by the small eddies is instead translated to a viscous transport. Turbulence dissipation usually dominates over viscous dissipation everywhere, except for in the viscous sublayer close to solid walls. Here, the turbulence model has to continuously reduce the turbulence level, such as in low Reynolds number models. Or, new boundary conditions have to be computed using wall functions.
The term “low Reynolds number model” sounds like a contradiction, since flows can only be turbulent if the Reynolds number is high enough. The notation “low Reynolds number” does not refer to the flow on a global scale, but to the region close to the wall where viscous effects dominate; i.e., the viscous sublayer in the figure above. A low Reynolds number model is a model that correctly reproduces the limiting behaviors of various flow quantities as the distance to the wall approaches zero. So, a low Reynolds number model must, for example, predict that k~y^{2} as y→0. Correct limiting behavior means that the turbulence model can be used to model the whole boundary layer, including the viscous sublayer and the buffer layer.
Most ω-based models are low Reynolds number models by construction. But the standard k-ε model and other commonly encountered k-ε models are not low Reynolds number models. Some of them can, however, be supplemented with so-called damping functions that give the correct limiting behavior. They are then known as low Reynolds number k-ε models.
Low Reynolds number models often give a very accurate description of the boundary layer. The sharp gradients close to walls do, however, require very high mesh resolutions and that, in turn, means that the high accuracy comes at a high computational cost. This is why alternative methods to model the flow close to walls are often employed for industrial applications.
The turbulent flow near a flat wall can be divided into four regions. At the wall, the fluid velocity is zero, and in a thin layer above this, the flow velocity is linear with distance from the wall. This region is called the viscous sublayer, or laminar sublayer. Further away from the wall is a region called the buffer layer. In the buffer region, turbulence stresses begin to dominate over viscous stresses and it eventually connects to a region where the flow is fully turbulent and the average flow velocity is related to the log of the distance to the wall. This is known as the log-law region. Even further away from the wall, the flow transitions to the free-stream region. The viscous and buffer layers are very thin and if the distance to the end of the buffer layer is , then the log-law region will extend about away from the wall.
It is possible to use a RANS model to compute the flow field in all four of these regions. However, since the thickness of the buffer layer is so small, it can be advantageous to use an approximation in this region. Wall functions ignore the flow field in the buffer region and analytically compute a nonzero fluid velocity at the wall. By using a wall function formulation, you assume an analytic solution for the flow in the viscous layer and the resultant models will have significantly lower computational requirements. This is a very useful approach for many practical engineering applications.
If you need a level of accuracy beyond what the wall function formulations provide, then you will want to consider a turbulence model that solves the entire flow regime as described for the low Reynolds number models above. For example, you may want to compute lift and drag on an object or compute the heat transfer between the fluid and the wall.
The automatic wall treatment functionality, which is new in COMSOL Multiphysics version 5.3, combines benefits from both wall functions and low Reynolds number models. Automatic wall treatment adapts the formulation to the mesh available in the model so that you get both robustness and accuracy. For instance, for a coarse boundary layer mesh, the feature will utilize a robust wall function formulation. However, for a dense boundary layer mesh, the automatic wall treatment will use a low Reynolds number formulation to resolve the velocity profile completely to the wall.
Going from a low Reynolds number formulation to a wall function formulation is a smooth transition. The software blends the two formulations in the boundary elements. Then, the software calculates the wall distance of the boundary elements’ grid points (this is in viscous units given by a liftoff). The combined formulations are then used for the boundary conditions.
All turbulence models in COMSOL Multiphysics, except the k-ε model, support automatic wall treatment. This means that the low Reynolds number models can be used for industrial applications and that their low Reynolds number modeling capability is only invoked when the mesh is fine enough.
The eight RANS turbulence models differ in how they model the flow close to walls, the number of additional variables solved for, and what these variables represent. All of these models augment the Navier-Stokes equations with an additional turbulence eddy viscosity term, but they differ in how it is computed.
The L-VEL and algebraic yPlus turbulence models compute the eddy viscosity using algebraic expressions based only on the local fluid velocity and the distance to the closest wall. They do not solve any additional transport equations. These models solve for the flow everywhere and are the most robust and least computationally intensive of the eight turbulence models. While they are generally the least accurate models, they do provide good approximations for internal flow, especially in electronic cooling applications.
The Spalart-Allmaras model adds a single additional variable for an undamped kinematic eddy viscosity. It is a low Reynolds number model and can resolve the entire flow field down to the solid wall. The model was originally developed for aerodynamics applications and is advantageous in that it is relatively robust and has moderate resolution requirements. Experience shows that this model does not accurately compute fields that exhibit shear flow, separated flow, or decaying turbulence. Its advantage is that it is quite stable and shows good convergence.
The k-ε model solves for two variables: k, the turbulence kinetic energy; and ε (epsilon), the rate of dissipation of turbulence kinetic energy. Wall functions are used in this model, so the flow in the buffer region is not simulated. The k-ε model has historically been very popular for industrial applications due to its good convergence rate and relatively low memory requirements. It does not very accurately compute flow fields that exhibit adverse pressure gradients, strong curvature to the flow, or jet flow. It does perform well for external flow problems around complex geometries. For example, the k-ε model can be used to solve for the airflow around a bluff body.
The turbulence models listed below are all more nonlinear than the k-ε model and they can often be difficult to converge unless a good initial guess is provided. The k-ε model can be used to provide a good initial guess. Just solve the model using the k-ε model and then use the new Generate New Turbulence Interface functionality, available in the CFD Module with COMSOL Multiphysics version 5.3.
The k-ω model is similar to the k-ε model, but it solves for ω (omega) — the specific rate of dissipation of kinetic energy. It is a low Reynolds number model, but it can also be used in conjunction with wall functions. It is more nonlinear, and thereby more difficult to converge than the k-ε model, and it is quite sensitive to the initial guess of the solution. The k-ω model is useful in many cases where the k-ε model is not accurate, such as internal flows, flows that exhibit strong curvature, separated flows, and jets. A good example of internal flow is flow through a pipe bend.
The low Reynolds number k-ε model is similar to the k-ε model, but does not need wall functions: it can solve for the flow everywhere. It is a logical extension of the k-ε model and shares many of its advantages, but generally requires a denser mesh; not only at walls, but everywhere its low Reynolds number properties kick in and dampen the turbulence. It can sometimes be useful to use the k-ε model to first compute a good initial condition for solving the low Reynolds number k-ε model. An alternative way is to use the automatic wall treatment and start with a coarse boundary layer mesh to get wall functions and then refine the boundary layer at the interesting walls to get the low Reynolds number models.
The low Reynolds number k-ε model can compute lift and drag forces and heat fluxes can be modeled with higher accuracy compared to the k-ε model. It has also shown to predict separation and reattachment quite well for a number of cases.
The SST model is a combination of the k-ε model in the free stream and the k-ω model near the walls. It is a low Reynolds number model and kind of the “go to” model for industrial applications. It has similar resolution requirements to the k-ω model and the low Reynolds number k-ε model, but its formulation eliminates some weaknesses displayed by pure k-ω and k-ε models. In a tutorial model example, the SST model solves for flow over a NACA 0012 Airfoil. The results are shown to compare well with experimental data.
Close to wall boundaries, the fluctuations of the velocity are usually much larger in the parallel directions to the wall in comparison with the direction perpendicular to the wall. The velocity fluctuations are said to be anisotropic. Further away from the wall, the fluctuations are of the same magnitude in all directions. The velocity fluctuations become isotropic.
The v2-f turbulence model describes the anisotropy of the turbulence intensity in the turbulent boundary layer using two new equations, in addition to the two equations for turbulence kinetic energy (k) and dissipation rate (ε). The first equation describes the transport of turbulent velocity fluctuations normal to the streamlines. The second equation accounts for nonlocal effects such as the wall-induced damping of the redistribution of turbulence kinetic energy between the normal and parallel directions.
You should use this model for enclosed flows over curved surfaces, for example, to model cyclones.
Solving for any kind of fluid flow problem — laminar or turbulent — is computationally intensive. Relatively fine meshes are required and there are many variables to solve for. Ideally, you would have a very fast computer with many gigabytes of RAM to solve such problems, but simulations can still take hours or days for larger 3D models. Therefore, we want to use as simple a mesh as possible, while still capturing all of the details of the flow.
Referring back to the figure at the top of this blog post, we can observe that for the flat plate (and for most flow problems), the velocity field changes quite slowly in the direction tangential to the wall, but quite rapidly in the normal direction, especially if we consider the buffer layer region. This observation motivates the use of a boundary layer mesh. Boundary layer meshes (which are the default mesh type on walls when using our physics-based meshing) insert thin rectangles in 2D or triangular prisms in 3D at the walls. These high-aspect-ratio elements will do a good job of resolving the variations in the flow speed normal to the boundary, while reducing the number of calculation points in the direction tangential to the boundary.
The boundary layer mesh (magenta) around an airfoil and the surrounding triangular mesh (cyan) for a 2D mesh.
The boundary layer mesh (magenta) around a bluff body and the surrounding tetrahedral mesh (cyan) for a 3D volumetric mesh.
Once you’ve used one of these turbulence models to solve your flow simulation, you will want to verify that the solution is accurate. Of course, as you do with any finite element model, you can simply run it with finer and finer meshes and observe how the solution changes with increasing mesh refinement. Once the solution does not change to within a value you find acceptable, your simulation can be considered converged with respect to the mesh. However, there are additional values you need to check when modeling turbulence.
When using wall function formulations, you will want to check the wall resolution viscous units (this plot is generated by default). This value tells you how far into the boundary layer your computational domain starts and should not be too large. You should consider refining your mesh in the wall normal direction if there are regions where the wall resolution exceeds several hundred. The second variable that you should check when using wall functions is the wall liftoff (in length units). This variable is related to the assumed thickness of the viscous layer and should be small relative to the surrounding dimensions of the geometry. If it is not, then you should refine the mesh in these regions as well.
The maximum wall liftoff in viscous units is less than 100, so there is no need to refine the boundary layer mesh.
When solving a low Reynolds number model without automatic wall treatment, check the dimensionless distance to cell center (also generated by default). This value should be of order unity everywhere for the algebraic models and less than 0.5 for all two-equation models and the v2-f model. If it is not, then refine the mesh in these regions.
In this blog post, we have discussed the various turbulence models available in COMSOL Multiphysics, highlighting when and why you should use each one of them. The real strength of the COMSOL® software is when you want to combine your fluid flow simulations with other physics, such as finding stresses on a solar panel in high winds, forced convection modeling in a heat exchanger, or mass transfer in a mixer, among other possibilities.
If you are interested in using the COMSOL® software for your CFD and multiphysics simulations, or if you have a question that isn’t addressed here, please contact us.
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A few years ago, we talked on the blog about how selective laser sintering was taking the 3D print world by storm. Since then, the popularity of this rapid prototyping technique has continued to grow throughout various industries. The same can be said of a closely related technique, selective laser melting, which uses a laser beam to melt powdered material in order to produce a 3D part.
A schematic describing the selective laser melting process. Image by Materialgeeza — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
Copper, aluminum, and stainless steel: these are just some metals that are already used in SLM. In recent years, researchers have experimented with adding high-melting materials into the mix. Molybdenum, shown below, is one example.
Molybdenum is a high-melting material with potential use in SLM. Image by Alchemist-hp — Own work. Licensed under Free Art License 1.3, via Wikimedia Commons.
With these new materials comes a new challenge: The processing window for refractory metals is significantly narrower. This means that further analysis is needed to gain an understanding of how high-melting materials behave during SLM. To address this multiphysics problem, researchers at Plansee SE in Austria turned to the COMSOL® software.
For their analysis, the researchers created a COMSOL Multiphysics model to analyze laser beam-matter interaction in SLM. The model geometry consists of a simple cubic metal powder layer, resting on top of a large base plate that is exposed to a Gaussian laser beam. Note that the model takes advantage of the symmetry in the direction that the laser moves.
The meshed model geometry. Image by K.-H. Leitz, P. Singer, A. Plankensteiner, B. Tabernig, H. Kestler, and L.S. Sigl and taken from their COMSOL Conference 2016 Munich paper.
To accurately model laser beam-matter interaction, there are several factors to account for:
Coupling thermal and fluid dynamics via the Heat Transfer Module and CFD Module enabled the researchers to investigate these factors. For this specific case, the angle dependency of absorption, shadowing effects, and various reflections are neglected.
In the analysis, the metal powder is represented by two different materials: stainless steel and molybdenum. The researchers compared the volume buildup at multiple stages of the SLM process for each material.
From the figures below, we can see a clear difference between the process dynamics for steel and molybdenum. In the case of steel, there is a long melt pool and significant effects via evaporation. In the case of molybdenum, the melt pool is confined to the size of the focal spot area and the temperatures are much lower than those causing evaporation. This difference can be traced back to the phase-transition temperatures and thermal conductivities for each material. Because of its high thermal conductivity, molybdenum experiences greater heat losses in SLM, which then restricts the melt pool size. These heat losses, in combination with molybdenum’s high evaporation temperature, prevent evaporation from occurring.
The volume buildup during the selective laser melting of steel (left) and molybdenum (right). Images by K.-H. Leitz, P. Singer, A. Plankensteiner, B. Tabernig, H. Kestler, and L.S. Sigl and taken from their COMSOL Conference 2016 Munich paper.
The above results provide a better understanding of the dynamics of SLM as well as the characteristics of the process that are specific to the material used. Since the core of the model describes laser beam-matter interaction, it can be used to study other manufacturing processes that involve lasers.
The new automatic wall treatment functionality in COMSOL Multiphysics version 5.3 combines the robustness of wall functions with the accuracy of low Reynolds number models by adapting the formulation to the mesh available in the model. If the boundary layer mesh is coarse, a robust wall function formulation is used. If the boundary layer mesh is dense, a low Reynolds number formulation is used, which resolves the velocity profile all the way to the wall.
The transition between the low Reynolds number formulation and the wall function formulation is smooth and is done by blending the two formulations in the boundary elements. By calculating the wall distance of the boundary elements’ grid points in viscous units given by a liftoff, the combination of the two formulations is used for the boundary conditions.
The figure below exemplifies the transition between the low Reynolds number formulation and the logarithmic wall functions for the low Re k-ε turbulence model. The wall distance in viscous units, y+, is plotted against the turbulence dissipation rate, ε. The green curve represents the low Re formulation of ε, the blue curve is the wall function representation, while the red curve is the Wolfshtein model that is used for the automatic wall treatment. Observe the smooth transition obtained with the Wolfshtein model (red) for y+ values ranging from 1 to 20; i.e., in the buffer layer.
Low Re formulation (green), wall functions (blue), and automatic wall treatment (red).
In order to verify the definition of a model, we can investigate how the walls are treated by plotting the y+ variable at the boundaries, as shown in the figure below. For this pipe elbow benchmark model, we can see that the low Reynolds number formulation dominates at the inner curved surface of the bend, while at the straight sections of the pipe, the wall function formulation dominates.
The deep red regions have a value of y+, or around 20, while the blue regions are at around 1.
The functionality for automatic wall treatment allows the use of low Reynolds number models for a wider range of problems. Examples are coupled problems where certain surfaces are subjected to flux of heat, chemical reactions, or fluid-structure interactions. Instead of having to use a dense mesh on all surfaces, which could be very computationally expensive, we can apply a dense mesh only on the relevant surfaces where we need to accurately resolve the boundary layer.
The figure below shows the boundary layer mesh for the solar panel model in the Application Library. We can see that the mesh on the surface of the panels is dense with tight boundary elements. On these surfaces, we need the forces exerted by the fluid on the structure with high accuracy in order to compute the stresses and strains as well as the displacements. The concrete base is not influenced by the forces of the wind and the forces on these surfaces do not require the same accuracy. The automatic wall treatment functionality allows for the solution of this problem by just clicking the Compute button, while a conventional low Reynolds number model would require a dense boundary layer mesh on all boundaries in order to converge.
In this fluid-structure interaction tutorial of a solar panel, we can use a coarse boundary layer mesh for the ground and the concrete foundation and a fine mesh on the surface of the panels, where the forces need to be calculated with accuracy.
The new functionality is also of great use in model development. In order to iron out the proper assumptions and boundary conditions, we may need quick results on a coarse mesh as a first step. Once we have verified our model formulation, we can refine the mesh in order to obtain more accurate results. The automatic wall treatment functionality allows for this type of model development with a minimum number of obstacles and operations: the “only” thing we need to do is to refine the mesh.
The robustness that this new functionality provides also simplifies the use of low Reynolds number turbulence models in general. A common procedure is to solve the model equations using wall functions with high Reynolds number models and then use this solution as the starting guess in the low Reynolds number models. The adaptive wall treatment reduces this procedure to the sequential solution of the model equations for a coarse and fine mesh; i.e., it eliminates the need for the formulation and solution of a high Reynolds number model as a first step.
Automatic wall treatment is available for all low Reynolds number turbulence model interfaces in the latest version of the CFD Module:
The automatic wall treatment functionality is available in the Settings windows for all of the above-mentioned flow interfaces. The figure below shows the selection for the Turbulent Flow, k-ω interface. In this flow interface, we can select from three different wall treatment options: automatic, wall functions, and low Reynolds number.
The three available options for the Turbulent Flow, k-ω interface: automatic, wall functions, and low Reynolds number.
Wind turbines (not to be confused with windmills) were first used to generate electrical power in the 19^{th} century. Scottish innovator James Blyth is largely credited with this invention. Supposedly, Blyth used a wind turbine to generate electricity for his cottage. He later improved the design to provide power to a nearby asylum.
Different types of VAWTs compared to the operation of a HAWT.
The use of wind-generated electrical power is continuously increasing. To help with the growing demand, engineers must ensure that wind turbine designs are highly efficient. This is a particular challenge for VAWTs, which struggle with peak efficiencies and starting torque, among other issues. On the other hand, VAWTs have several benefits. For instance, they are easier to install and maintain than HAWTs and don’t need to point windward. VAWTs can also be used in urban areas and in smaller scales and are a low risk for humans and wildlife.
To overcome the shortcomings of VAWTs while harnessing their advantages, we can use pitch control systems. These systems improve the starting torque and efficiency and provide a broader operating range than traditional fixed-pitch VAWTs.
Let’s explore simulation research from a team at Sam Houston State University and Southeastern Louisiana University that examines how airfoil pitch control affects the efficiency of a VAWT.
The researchers investigated a dynamic control system that contains both pitch and camber controls. This system is comprised of three blades (or airfoils) with flaps at the trailing edge. At the center of the system is a vertical-axis hub that is attached to blade-supporting arms. Each airfoil is connected to one of these support arms and freely pivots with respect to the arm. The independent pivoting enables the airfoils to have adjustable wind attack angles.
With an airfoil pitch control system, the angle between the support arms and airfoils can be altered for any given support arm position. This changes the attack angles between the wind and airfoils, redistributing the air pressure on the airfoil surfaces.
Top view of a VAWT system with both pitch and camber controls. Image by J. Ma, C. Koutsougeras, and H. Luo and taken from their COMSOL Conference 2016 Boston presentation.
From this system, the researchers modeled a 2D NACA 0012 airfoil within a rectangular area that represents one section of a wind tunnel. To simulate rotation, the rectangular box is moved while the airfoil is fixed in the horizontal direction.
Left: A NACA 0012 airfoil inside a rectangular wind tunnel. Right: The rectangular box is rotated to simulate an airfoil rotated 30°. The mesh seen here is used for illustration purposes. Images by J. Ma, C. Koutsougeras, and H. Luo and taken from their COMSOL Conference 2016 Boston paper.
The researchers used this model to analyze velocity and pressure at different airfoil pivot angles (wind angles of attack), support arm angles, and wind speeds. Through this, they can determine an optimal pivot angle for any given support arm position, maximizing torque and energy production and achieving a higher efficiency than VAWTs with fixed airfoils.
It’s important to note that this analysis does not include the centrifugal and Coriolis forces found in a VAWT system. As such, this study may be a good start, but the team needs to account for the influence of inertial forces for accurate and proper optimization.
The research team studied their model’s velocity profile at a 30° angle of attack when exposed to a wind speed of 20 m/s. They found the maximum air flow speed (35.6 m/s) near the leading edge of the airfoil and the minimum air flow speed (0.02 m/s) at the leading edge on the bottom side of the airfoil. This air flow behavior is mirrored when studying different airfoil rotation angles.
Left: The velocity field at a 30° angle of attack. Right: The velocity on the top and bottom airfoil surfaces. Images by J. Ma, C. Koutsougeras, and H. Luo and taken from their COMSOL Conference 2016 Boston presentation.
When looking at the pressure field of the airfoil surface at the same 30° angle of attack, the team noted that the peak pressure occurs in areas with the lowest flow speeds (close to the leading edge at the top of the airfoil) and vice versa. At the bottom of the airfoil, velocity and pressure remain almost constant except for the change near the leading edge.
Left: The pressure field at a 30° angle of attack. Right: The pressure on the top and bottom airfoil surfaces. Images by J. Ma, C. Koutsougeras, and H. Luo and taken from their COMSOL Conference 2016 Boston presentation.
Moving on, let’s examine the results for torque. Here, the researchers calculated torque in relation to the VAWT’s rotation center from the pressure field on the airfoil surface. As we can see in the following plot, torque increases with the angle of attack until it reaches its peak value at a 90° angle of attack. After this, torque drops and eventually returns to near-initial values as the angle of attack continues to grow.
The torque with a support arm angle of 30° and wind speed of 20 m/s. Image by J. Ma, C. Koutsougeras, and H. Luo and taken from their COMSOL Conference 2016 Boston presentation.
Next, let’s see how torque values are affected by other support arm angles. The following results indicate that torque peaks at a 90° angle of attack for all support arm angles. As such, designing a control mechanism that keeps NACA 0012 airfoils at a constant 90° angle of attack will maximize device efficiency and energy production.
Comparisons of the torque distribution and attack angles for various support arm angles. The different curves represent support arm angles ranging from 0° to 90° (left) and 90° to 180° (right) at 10° increments. Images by J. Ma, C. Koutsougeras, and H. Luo and taken from their COMSOL Conference 2016 Boston paper.
In regards to laminar and turbulent wind speeds, when the support arm is constantly at 30°, there is an insignificant gain in torque due to the increased wind speed. Further, wind speed does not affect the torque distribution pattern.
Comparison of the effects of different wind speeds. Image by J. Ma, C. Koutsougeras, and H. Luo and taken from their COMSOL Conference 2016 Boston presentation.
Using the knowledge gained from these studies, the researchers found that there is an optimal airfoil pivot angle that yields maximum torque at a given VAWT rotation angle. This information provides guidance for designing airfoil pitch control systems. For their next step, the team can increase the accuracy of their optimization by including centrifugal and Coriolis forces. Like we mentioned before, this is an important step in improving the study.
The researchers also hope to improve their studies by accounting for other factors, including multiple airfoils and support arms as well as the effect of wake from wind passing through one airfoil to another. In future studies, they are considering using a 2D and 3D multiblade model, studying a nonsymmetric airfoil, and more.
Building physics engineers aim to improve the energy performance and sustainability of building envelopes. Although their practices are based on past experience, new materials and building techniques are constantly being developed that offer a wide set of options in building design and thermal management. Let’s see how to model heat and moisture transport in building materials to help reduce energy costs and preserve buildings.
Building envelopes can be analyzed by modeling heat and moisture transport.
Controlling moisture is necessary to optimize the thermal performance of building envelopes and reduce energy costs. The thermal properties of insulation or isolation materials usually depend on both temperature and moisture content. Therefore, a coupled heat and moisture model helps us fully analyze the thermal performance of a building component. One example is the dependence of a lime silica brick’s thermal conductivity on relative humidity.
The moisture dependence of thermal conductivity for lime silica brick.
The figure above shows that lime silica brick becomes two times less thermally isolating for high relative humidity values.
In addition, we must consider moisture control in the building design process to choose building components that can reduce the risk of condensation. The coupled modeling of heat and moisture transport enables us to analyze different moisture variations and phenomena in building components, such as:
Let’s consider a wood-frame wall between a warm indoor environment and a cold outdoor environment. Vapor diffuses through the wall from the high-moisture environment inside to the low-moisture environment outside. This creates high relative humidity values associated with high temperature values close to the exterior panel, with the risk of condensation as a direct consequence.
The relative humidity distribution in a wood-frame wall.
Condensation leads to mold growth, which directly affects human health and building sustainability. The rate of mold growth is key data for the preservation of historical buildings, for example. To prevent the risk of interstitial condensation, it is common practice to add a vapor barrier between the interior gypsum panel and the cellulose isolation board. This reduces the moisture values where they are at a maximum. The figure below shows the relative humidity distribution across the wood-frame wall through a wood stud (red lines) and a cellulose board (blue lines), with and without the vapor barrier (dashed lines and solid lines, respectively).
Effect of a vapor barrier on relative humidity distribution across the wood-frame wall in a wood stud and cellulose board.
For this model, we consider the building materials to be specific unsaturated porous media in which the moisture exists in both liquid and vapor phases and only some transport processes are relevant. The norm EN 15026 standard addresses the transport moisture phenomena that is taken into account in building materials, following the theory expressed in Ref. 1.
The transport equation established as a standard by the norm accounts for liquid transport by capillary forces, vapor diffusion due to a vapor pressure gradient, and moisture storage.
We model the latent heat effect due to vapor condensation by adding the following flux in the heat transfer equation:
In addition, the moisture dependence of the thermal properties is assessed.
Find details about the moisture transport equation in building materials in the Heat Transfer Module User’s Guide.
When using the Heat Transfer Module, the Heat and Moisture Transport interface adds a:
Finally, the latent heat source due to evaporation is added to the heat transfer equation by the Building Material feature of the Heat Transfer interface.
The model tree and subsequent subnodes when choosing the Heat Transfer in Building Materials interface, along with the Settings window of the Building Material feature.
Modeling heat and moisture transport in an unsaturated porous medium is important for analyzing polymer materials for the pharmaceutical industry, protective layers on electrical cables, and food-drying processes, to name a few examples.
For these applications, phenomenological models, such as the one presented above for building materials, may not be available. However, by considering the conservation of heat and moisture in each phase (solid, liquid, and gas), and volume averaging over the different phases, we can derive a mechanistic model.
To compute the moisture distribution, we solve a two-phase flow problem in the porous medium. Two equations of transport are solved: one for the vapor and one for the liquid water. The coupling between the vapor and liquid water operates through the definition of saturation variables, S_{vapor} + S_{liquid} = 1. The changing water saturation is taken into consideration for the definition of the effective vapor diffusivity and liquid permeability.
For quick processes, with a time scale comparable to the time it takes to reach equilibrium between the liquid and gas phases inside the pores of the medium, a nonequilibrium formulation can be defined through the following evaporation flux:
In this definition, the equilibrium vapor concentration, defined as the product of the saturation concentration c_{sat} and the water activity a_{w}, is used to account for the porous medium structure. Indeed, due to capillary forces, equilibrium is reached for concentrations that are lower than in a free medium.
By letting the evaporation rate K go to infinity, an equilibrium formulation is obtained with the vapor concentration equal to the equilibrium concentration.
Let’s consider a food-drying process. A piece of potato, initially saturated with liquid water, is placed in an airflow to be dried. Inside the potato, the vapor is transported by binary diffusion in air. We use a Brinkman formulation to model the flow induced by the moist air pressure gradient in the pores. As the liquid phase velocity is small compared to the moist air velocity, Darcy’s law is used for the liquid water flow due to the pressure gradient. The capillary flow, due to the difference between the relative attraction of the water molecules for each other and the potato, is also considered in the liquid water transport.
The vapor and liquid water distributions over time for this model are shown in the following two animations. Note that water can leave the potato as vapor only.
The liquid water concentration over time.
The vapor is transported away by the airflow, as shown in this animation:
The water vapor concentration over time.
The evaporation causes a reduction of the temperature in the potato. The temperature distribution over time is shown below.
Temperature distribution over time.
You can implement the equations in the Heat Transfer in Porous Media interface within the Heat Transfer Module and the Transport of Diluted Species interface within the Chemical Reaction Engineering Module. This process requires some steps in order to couple the multiphase flow in a porous medium together with the evaporation process.
Read the article “Engineering Perfect Puffed Snacks” on pages 7–9 of COMSOL News 2017 to see how Cornell University researchers used COMSOL Multiphysics to model rice puffing. In this numerically challenging process, the rapid evaporation of liquid water results in a large gas pressure buildup and phase transformation in the grain.
In this blog post, we discussed COMSOL® software features for modeling heat and moisture transport in porous media. COMSOL Multiphysics (along with the Chemical Reaction Engineering Module and Heat Transfer Module) provides you with tools to define the corresponding phenomenological and mechanistic models for a large range of applications. Depending on the dominant transport processes, you can use predefined interfaces or define your own model.
Künzel, H. 1995. Simultaneous Heat and Moisture Transport in Building Components. One and two-dimensional calculation using simple parameters. PhD Thesis. Fraunhofer Institute of Building Physics.
Solar-grade silicon is one of three grades of high-purity silicon. Each grade has different applications and specific purity percentage requirements:
The structure of monocrystalline silicon. Solar-grade silicon is almost pure silicon.
Traditionally, solar-grade silicon is produced using high temperatures (2000°C) to reduce silicon quartz and carbon, resulting in silicon with a 98.5% purity. This isn’t quite pure enough to be considered solar grade, so the silicon must be refined further through a gas phase. With multiple steps and different processes, this method isn’t efficient. It is also energy intensive, expensive, and requires experienced operators.
The method that JPM analyzed starts with raw materials that are highly pure. The silicon is placed into a contaminant-free microwave oven that performs both the heating and gas phase stages of the traditional production process. Since there’s no consecutive refinement processes, this approach is more efficient and cost effective.
The setup for the microwave furnace. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.
The microwave furnace consists of five parts:
One advantage of an optimized microwave furnace design is that there is reduced heat loss. This is partially due to the selective heating, which heats materials on a volumetric heat input, leading to a temperature drop from the inside out. In addition, there’s less diffusion of the silicon’s impurities because the furnace has a faster warming time and shorter residence time.
To optimize the microwave furnace for solar-grade silicon production, JPM Silicon GmbH studied its internal processes with the COMSOL Multiphysics® software.
The research team set up their model to include the electromagnetic, chemical, and physical phenomena occurring within the microwave furnace. Since some materials have electromagnetic properties that are strongly temperature dependent, the model couples the electromagnetic field distribution and temperature field.
You can learn more about the model setup by reading the full conference paper “Multiphysics Modelling of a Microwave Furnace for Efficient Solar Silicon Production“.
It’s important to use chemically stable structural materials and an inert gas in the microwave furnace to avoid unwanted reactions. Also, the insulation materials must be effective in minimizing heat losses.
The research team used the RF Module to simulate the electromagnetic intensity and distribution in the resonator and silicon sample. They used Maxwell’s equations to determine the propagation of the microwave radiation.
The electric field is higher at the height of the waveguide ports than at any other part of the reaction chamber. The field enhancement in the crucible’s core indicates that this is the optimal location for the crucible to be heated, as shown in the results below.
The distribution of the electric field in the resonator and waveguide. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.
The researchers also wanted to see how varying the height of the insulation plate affects the operation of the furnace. They tested three different heights for the plate (which the crucible sits on top of) and reexamined the electric field. The different insulation plate heights include:
The distribution of the electric field when the height of the insulation plate is 30 mm (left), 40 mm (middle), and 50 mm (right). Images by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich paper.
The simulation results show that the 40-mm insulation plate performs best. The electric field is focused at the center of the crucible, thus on the silicon sample.
The CFD Module solves for the Navier-Stokes equations, allowing the researchers to find the gas flow velocity distribution. The gas flows from the inlet over the surface of the silicon sample, rather than having a homogeneous velocity. The wall then deflects the flow toward the outlet. The simulation shows that only a slight gas flow exists near the waveguide ports as well as near the top and bottom walls.
The distribution of gas velocity in the resonator. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich poster.
To analyze how well the electromagnetic waves heat the silicon sample, the research team examined the heat distribution in the resonator. Their model includes forced heat equations to calculate conduction, convection, and radiation from solids and liquids (Planck’s radiation law) as well as gases (Stefan-Boltzmann law). The dissipated heat, solved with the RF Module, is used as a volumetric heat source. The gas velocity profile, calculated with the CFD Module, helps find the convective thermal losses.
As expected from the electromagnetics study, the hottest point in the resonator is at the crucible’s core. Further, the surrounding insulation layers don’t heat up as much, thanks to their lower thermal conductivity.
The distribution of heat in the resonator. Image by N. Rezaii and J.P. Mai and taken from their COMSOL Conference 2016 Munich poster.
By gaining insight into the internal processes of a microwave furnace, researchers from JPM Silicon GmbH were able to optimize their design and pave the way for efficient solar-grade silicon production.
In the 1960s, G. Segré and A. Silberberg observed a surprising effect: When carried through a laminar pipe flow, neutrally buoyant particles congregate in a ring-like structure with a radius of about 0.6 times the pipe radius. This correlates to a distance from the parallel walls of around 0.2 times the width of the flow channel. The reason for this behavior, as they would discover decades later, could be traced back to the forces that act on particles in an inertial flow.
Today, we use the term inertial focusing to describe the migration of particles to a position of equilibrium. This technique is widely used in clinical and point-of-care diagnostics as a way to concentrate and isolate particles of different sizes for further analysis and testing.
Many types of medical diagnostics use inertial focusing for testing and analysis. Image in the public domain, via Wikimedia Commons.
In order for inertial focusing to be effective in these and other applications, accurately analyzing the migration patterns of the particles is a key step. A new benchmark example from the latest version of COMSOL Multiphysics — version 5.3 — highlights why the COMSOL® software is the right tool for obtaining reliable results.
For this example, we consider the particle trajectories in a 2D Poiseuille flow. To account for relevant forces, we use derived expressions from a similar migration of particles in a 2D parabolic flow inside of two parallel walls (see Ref. 2 in the model documentation). Built-in corrections for both the lift and drag forces allow us to account for the presence of these walls in the simulation analysis.
Note: Lift and drag forces make up the total force acting on neutrally buoyant particles inside a creeping flow. By definition, the gravitational and buoyant forces cancel out one another.
We assume that the lift force acts only perpendicular to the direction of the fluid velocity. It is also assumed that the spherical particles are small in comparison to the width of the channel and that they are rotationally rigid.
To compute the velocity field, we use the Laminar Flow physics interface. This is then coupled to the Particle Tracing for Fluid Flow interface via the Drag Force node. Thanks to the Laminar Inflow boundary condition, we can automatically compute the complete velocity profile at the inlet boundary. For the laminar flow of a Newtonian fluid inside two parallel walls, it is known that the profile will be parabolic. This means that we could have directly entered the analytic expressions for fluid velocity. However, we opt to use the Laminar Flow physics interface in this case, as it demonstrates the workflow that is most appropriate for a general geometry.
Now let’s move on to the results. First, we can look at the fluid velocity magnitude in the channel. As expected, the velocity profile is parabolic. Note that the aspect ratio of the geometry is 1000:1, so the channel is very long compared to its height. The plot uses an automatic view scale to make the results easier to visualize.
The parabolic fluid velocity profile within a channel that is bound by two parallel walls.
We can then shift our attention to the trajectories of the neutrally buoyant particles. Note that in the plot below, the color expression represents the y-component of the particle velocity in mm/s. The results indicate that all of the particles are close to equilibrium positions at distances of about 0.3 D on either side of the center of the channel. (D represents the width of the channel). It does, however, take longer for particles released near the center of the channel to reach these positions. Their initial force is weaker as they are released in the area where the velocity gradient is smallest. From the plots, we can see that the particles converge at heights that are 0.2 and 0.8 times the width of the channel. These findings show good agreement with experimental observations.
The trajectory of particles inside the channel.
The last two plots show the average and standard deviation of the normalized distance between the particles and the center of the channel. These results verify that the equilibrium distance from the center of the channel is in fact around 0.3 D.
The average (left) and standard deviation (right) of the normalized distance between the particles and center of the channel.
In order to effectively use inertial focusing for medical and other applications, you need to first understand the behavior of particles as they migrate through a channel to positions of equilibrium. With COMSOL Multiphysics® version 5.3, you can perform these studies and generate reliable results. This accurate description of inertial focusing serves as a foundation for analyzing and optimizing designs that rely on this technique.
Now it’s your turn! Give our new benchmark model a try:
Interested in learning about further updates in version 5.3 of COMSOL Multiphysics? You can get the full scoop in our 5.3 Release Highlights.
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The transport of momentum normal to the boundary in turbulent boundary layers is strongly damped by the presence of the solid wall. Also, the normal component of the turbulence intensity is more strongly suppressed by the proximity to the solid wall than the in-plane components. This means that mass and heat transfer in the normal direction to the wall are also partly blocked. Traditionally, this blocking effect has been dealt with by introducing damping functions for the normal component of the turbulent viscosity in wall-resolved turbulence models.
Using direct numerical simulations, Durbin confirmed that the near-wall damping of the eddy viscosity is caused by the suppression of the turbulence intensity’s normal component according to the following relation:
(1)
where C_{μ} denotes a model constant, is the variance of the normal component of the turbulent velocity, and T represents the turbulence time scale.
The effect of the wall on the damping of the turbulence intensity is depicted in the figure below. The distance from the surface of a bubble or platelet-shaped object from their centers represents an average of the turbulence intensity at the centerpoint of the bubble or platelet shape.
Close to the walls, the fluctuations of the velocity are much larger in the x direction and y direction in comparison with the z direction. The velocity fluctuations are anisotropic and the surface takes the shape of a platelet. Further away from the wall, the fluctuations are of the same magnitude in all three dimensions: x, y, and z. The velocity fluctuations are isotropic and the surface takes the shape of a spherical bubble. The relative size of the bubbles and platelet shapes represents the relative size of the fluctuating eddies in the different directions.
A representation of the average velocity fluctuations as imaginary bubbles or platelet shapes. The distance from the surface to the center of the bubbles or platelets represents the average magnitude of the turbulence intensity.
The v2-f turbulence model describes the anisotropy of the turbulence intensity in the turbulent boundary layer using two new equations, in addition to the two equations for turbulence kinetic energy and dissipation rate. The implementation used in the CFD Module, an add-on product to COMSOL Multiphysics, is based upon the formulation by F. Billard, J. Uribe, and D. Laurence from their 2008 paper.
To accurately describe the transport of turbulence energy and the redistribution of the turbulence energy in different directions, the v2-f model introduces two new equations. The first equation describes the transport of turbulence normal to the wall using a variable that is equal to (usually denoted φ or ς in scientific literature). In this equation, denotes the variance of the normal component of the turbulent velocity and k denotes the turbulence kinetic energy.
The second equation is an elliptic partial differential equation for the blending coefficient (usually denoted α in scientific literature). The elliptic blending equation accounts for nonlocal effects such as the wall-induced damping of the redistribution of turbulence kinetic energy between the normal and parallel directions. Due to the name of the independent variables, this model is often referred to as the φ-α version of the v2-f turbulence model. The model is equally as robust as standard isotropic turbulence models and as accurate as the original, less robust formulation of the v2-f model.
The implementation in the CFD Module has been verified using a few benchmark cases, including the periodic hill benchmark. In this model, a flow is forced over two parallel hills where the velocity field is periodic on the two vertical inlet and outlet boundaries (shown in the figure below). This means that while there is a pressure drop from the inlet to the outlet boundary, their velocity fields are identical.
In the figure below, the flow is from left to right. We can see that there is a detachment of the flow with a formation of a recirculation zone behind the hills in the direction of the flow. The size of the recirculation zone and the distance from the inlet to the reattachment of the flow is inherently difficult to predict using RANS turbulence models. The results obtained with the implementation of the CFD Module are in excellent agreement with the results reported in scientific literature.
Flow over two hills where the velocity profiles at the vertical boundaries above the two hilltops are identical for the inlet and outlet boundaries. The flow is from left to right.
Another interesting problem is the flow in a hydrocyclone with two tangential inlets. The two outlets are at the top and bottom. The stream from the bottom usually contains the undesired particles, thus it is called the reject stream. The outlet stream at the top is referred to as the accept stream. The difficulty is to capture the semifree vortex, which is unattainable for standard two-equation models.
For more information about hydrocyclone simulations, see this research paper.
Left: Velocity field (streamlines) and pressure field (cross-section plot) in a hydrocyclone. Right: Isosurface of the absolute value of the vorticity in the cyclone and pressure cross section. The simulation captures the free vortex in the center of the cyclone.
The figure below shows the azimuthal component of the velocity just below the top outlet. The profile with a maximum at a radial position just outside of the outlet pipe is in good agreement with the results reported in scientific literature. In addition, the decrease of the azimuthal velocity away from the maximum outward along the radius is in good agreement with the literature.
Azimuthal component of the velocity as a function of the radius in the cyclone just below its horizontal top outlet.
To summarize, the new v2-f turbulence model widens the applicability of the CFD Module to include cases that require anisotropic turbulence modeling. This model gives an accuracy that is not possible to obtain with two-equation models, yet it is as robust as standard two-equation models.
Let’s start by considering a model of the electrical heating of a busbar, shown below. You may recognize this as an introductory example to COMSOL Multiphysics, but if you haven’t already modeled it, we encourage you to review this model by going through the Introduction to COMSOL Multiphysics PDF booklet.
Electric currents (arrow plot) flowing through a metal busbar lead to resistive heating that raises the temperature (color surface plot).
In this example, we model electric current flowing through a busbar. This leads to resistive heating, which in turn causes the temperature of the busbar to rise. We assume that there is only heat transfer to the surrounding air, neglecting any conductive heat transfer through the bolts and radiative heat transfer. The example also initially assumes that there isn’t any fan forcing air over the busbar. Thus, the transfer of heat to the air is via natural, or free, convection.
As the part heats the surrounding air, the air gets hotter. As the air gets hotter, its density decreases, causing the hot air to rise relative to the cooler surrounding air. These free convective air currents increase the rate of heat transfer from the part to the surrounding air. The air currents depend on the temperature variations as well as the geometry of the part and its surroundings. Convection can, of course, also happen in any other gas or liquid, such as water or transformer oil, but we will center this discussion primarily around convection in air.
We can classify the surrounding airspace into one of two categories: Internal or External. Internal means that there is a finite-sized cavity (such as an electrical junction box) around the part within which the air is reasonably well contained, although it might have known air inlets and outlets to an external space. We then assume that the thermal boundary conditions on the outside of the cavity and at the inlets and outlets are known. On the other hand, External implies that the object is surrounded by what is essentially an infinitely large volume of air. We then assume that the air temperature far away from the object is a constant, known value.
The settings for a constant heat transfer coefficient.
The introductory busbar example assumes free convective heat transfer to an external airspace. This is modeled using the following boundary condition for the heat flux:
where the external air temperature is T_{ext} = 25°C and is the heat transfer coefficient.
This single-valued heat transfer coefficient represents an approximate and average of all of the local variations in air currents. Even for this simple system, any value between could be an appropriate heat transfer coefficient, and it’s worth trying out the bounding cases and comparing results.
If we instead know that there is a fan blowing air over this structure, then due to the faster air currents, we use a heat transfer coefficient of to represent the enhanced heat transfer.
If the surrounding fluid is a liquid such as water, then the range of free and forced heat transfer coefficients are much wider. For free convection in a liquid, is the typical range. For forced convection, the range is even wider: .
Clearly, entering a single-valued heat transfer coefficient for free or forced convection is an oversimplification, so why do we do it? First, it is simple to implement and easy to compare the best and worst cases. Also, this boundary condition can be applied with the core COMSOL Multiphysics package. However, there are some more sophisticated approaches available within the Heat Transfer Module and CFD Module, so let’s look at those next.
A convective correlation is an empirical relationship that has been developed for common geometries. When using the Heat Transfer Module or CFD Module, these correlations are available within the Heat Flux boundary condition, shown in the screenshot below.
The Heat Flux boundary condition with the external natural convection correlation for a vertical wall.
Using these correlations requires that you enter the part’s characteristic dimensions. For example, with our busbar model, we use the External natural convection, Vertical wall correlation and choose a wall height of 10 cm to model the free convective heat flux off of the busbar’s vertical faces. We also need to specify the external air temperature and pressure. These values can be loaded from the ASHRAE database, a process we describe in a previous blog post.
The table below shows schematics for all of the available correlations. They take the information about the surface geometry and use a Nusselt number correlation to compute a heat transfer coefficient. For the horizontally aligned faces of the busbar, for example, we use the Horizontal plate, Upside and Horizontal plate, Downside correlations.
When using the Forced Convection correlations, you must also enter the air velocity. These convective correlations have the advantage of being a more accurate representation of reality, since they are based on well-established experimental data. These correlations lead to a nonlinear boundary condition, but this usually results in only slightly longer computation times than when using a constant heat transfer coefficient. The disadvantage is that they are only appropriate to use when there is an empirical relationship that is reasonable for the part geometry.
Free Convection | Forced Convection | |
---|---|---|
External | ||
Internal |
The available Convective Correlation boundary conditions.
Note that all of the above convective correlations, even those classified as Internal, assume the presence of an infinite external reservoir of fluid; e.g., the ambient airspace. The heat carried away from the surfaces goes into this ambient airspace without changing its temperature, and the ambient air coming in is at a known temperature. If, however, we are dealing with convection in a completely enclosed container, then none of these correlations are appropriate and we must move to a different modeling approach.
Let’s consider a rectangular air-filled cavity. If this cavity is heated on one of the vertical sides and cooled on the other, then there will be a regular circulation of the air. Similarly, there will be air circulation if the cavity is heated from below and cooled from above. These cases are shown in the images below, which were generated by solving for both the temperature distribution and the air flow.
Free convective currents in vertically and horizontally aligned rectangular cavities.
Solving for the free convective currents is fairly involved. See, for example, this blog post on modeling natural convection. Therefore, we might like to find a simpler alternative. Within the Heat Transfer Module, there is the option to use the Equivalent conductivity for convection feature. When using this feature, the effective thermal conductivity of the air is increased based upon correlations for the horizontal and vertical rectangular cavity cases, as shown in the screenshot below.
The Equivalent conductivity for convection feature and settings.
The air domain is still explicitly modeled using the Fluid domain feature within the Heat Transfer interface, but the air flow fields are not computed and the velocity term is simply neglected. The thermal conductivity is increased by an empirical correlation factor that depends on the cavity dimensions and the temperature variation across the cavity. The dimensions of the cavity must be entered, but the software can automatically determine and update the temperature difference across the cavity.
Temperature distribution in vertically and horizontally aligned cavities using the Equivalent conductivity for convection feature. The free convective air currents are not computed. Instead, the thermal conductivity of the air is increased.
This approach for approximating free convection in a completely closed cavity requires us to mesh the air domain and solve for the temperature field in the air, but this usually adds only a small computational cost. The disadvantage of this approach is that it is not very applicable for nonrectangular geometries.
Next, let’s consider a completely sealed enclosure, but with a fan or blower inside that actively mixes the air. We can reasonably assume that well-mixed air is at a constant temperature throughout the cavity. In this case, it is appropriate to use the Isothermal Domain feature, which is available with the Heat Transfer Module when the Isothermal domain option is selected in the Settings window.
The settings associated with using the Isothermal Domain interface.
A well-mixed air domain can be explicitly modeled using the Isothermal Domain feature. In the model, the temperature of the entire domain is a constant value. The temperature of the air is computed based upon the balance of heat entering and leaving the domain via the boundaries. The Isothermal Domain boundaries can be set as one of the following options:
Of all of these boundary condition options, the Convective Heat Flux is the most appropriate for well-mixed air in an enclosed cavity.
Representative results when using an Isothermal Domain feature. The well-mixed air domain is a constant temperature and there is heat transfer to the surrounding solid domains via a specified heat transfer coefficient.
The most computationally expensive approach, but also the most general, is to explicitly model the airflow. We can model both forced and free convection as well as simulate an internal or external flow. This type of modeling can be done with either the Heat Transfer Module or CFD Module.
An example of computing air flow and temperature within an enclosure.
If you finished the Introduction to COMSOL Multiphysics booklet, you have already solved one example of an internal forced convection model. You can learn more about explicitly modeling airflow in the resources mentioned at the end of this post.
We will finish up this topic by addressing the question: When can free convection in air be ignored and how can we model these cases? When a cavity’s dimensions are very small, such as a thin gap between parts or a very thin tube, we run into the possibility that the viscous damping will exceed any buoyancy forces. This balance of viscous to buoyancy forces is characterized by the nondimensional Rayleigh number. The onset of free convection can be quite varied depending on boundary conditions and geometry. A good rule of thumb is that for dimensions less than 1mm, there will likely not be any free convection, but once the dimensions of the cavity get larger than 1cm, there likely will be free convective currents.
So how can we model heat transfer through these small gaps? If there is no air flow, then these air-filled regions can simply be modeled as either a solid or a fluid with no convective term. This is demonstrated in the Window and Glazing Thermal Performances tutorial. It is also appropriate to model the air as a solid within any microscale enclosed structure.
If these thin gaps are very small compared to the other dimensions of the system being analyzed, you can further simplify the gaps by modeling them via the Thin Layer boundary condition with a Thermally thick approximation layer type. This boundary condition introduces a jump in temperature across interior boundaries based on the specified thickness and thermal conductivity.
The Thin Layer boundary condition can model a thin air gap between parts.
We can use the previous two approaches within the core COMSOL Multiphysics package. In the Heat Transfer Module, there are additional options for the Thin Layer condition to consider more general and multilayer boundaries, which can be composed of several layers of materials.
Before closing out this discussion, we should also quickly address the question of radiative heat transfer. Although we haven’t discussed radiation here, an engineer must always take it into consideration. Surfaces exposed to ambient conditions will radiate heat to the surroundings and be heated by the sun. The magnitude of radiative heating from the sun is significant — about 1000 watts per square meter — and should not be neglected. For details on modeling radiative heat transfer to ambient conditions, read this previous blog post.
There will also be radiative heat transfer between interior surfaces. Radiative heat flux between surfaces is a function of the difference of temperature to the fourth power. Keep in mind that radiative heat transfer between two surfaces at 20°C and 50°C will be 200 watts per square meter at most, but rises to 1000 watts per square meter for surfaces at 20°C and 125°C. To correctly compute the radiative heat transfer between surfaces, it is also important to compute the view factors with the Heat Transfer Module.
Today we looked at several approaches for modeling convection, starting from the simplest approach of using a constant convective heat transfer coefficient. We then discussed using an Empirical Convective Correlation boundary condition before going over how to use an effective thermal conductivity within a domain and an isothermal domain feature, approaches with higher accuracy and only a slightly greater computational cost. The most computationally intensive approach — explicitly computing the flow field — is, of course, the most general. We also touched on when it is appropriate to neglect free convection entirely and how to model such situations. You should now have a greater understanding of the available options and trade-offs for modeling free and forced convection. Happy modeling!