Modeling subsurface flow problems often deals with large modeling domains that are exposed to sources or sinks of a comparatively small size.
Previously, wells were introduced in COMSOL Multiphysics as small 3D cylinders in a large layered subsurface domain, as shown in this blog post about coupling heat transfer and porous media flow in a geothermal doublet. This approach was necessary to apply suitable boundary conditions and involved meshing small objects.
With the Well boundary condition — new in COMSOL Multiphysics 5.3 — the cylinders can be replaced by edges, which is better for the meshing algorithm and requires less meshing. The highlight of this feature is that it provides accurate solutions compared to resolving details explicitly. Let’s discuss the Well boundary condition in more detail.
We first want to get familiar with the new boundary condition and its settings. The Well boundary condition is available as a Point feature in 2D and an Edge feature in 3D and can be used with the Darcy’s Law, Richards’ Equation and Two-Phase Darcy’s Law interfaces. With this boundary condition, you can choose whether the well is an injection or production well and specify either the pressure or mass flow. The images below show some of the different options that are available.
The settings for the modeling of an injection well with the Darcy’s Law, Richards’ Equation interface (left) and the Two-Phase Darcy’s Law interface (right), where the saturation must also be specified.
Let’s now see how the Well boundary condition compares with other options for modeling wells. For illustration purposes, we use a basic model, as shown in the image below.
Model geometry of a well with a radius of 0.5 m in a reservoir that has a radius of 20 m, height of 3 m, and is surrounded by an infinite elements domain.
Infinite elements are used so that we can apply a pressure at a large distance from the well without increasing the modeling domain. The geometry shown here resolves the well as a cylindrical surface. To be able to apply a boundary condition, the well cylinder must be cut out of the reservoir. Alternatively, a Mass Flux edge condition can be used — but only if we want to apply a mass flux, not a pressure. As of COMSOL Multiphysics 5.3, the Well boundary condition is available, which is suitable for pressure as well as mass flux conditions.
We compare the mesh in the two cases using exactly the same mesh settings. Here, we have 65,674 domain elements for the case where the well is fully resolved versus 28,728 domain elements when using the Well boundary condition. That is less than half the number of mesh elements.
Comparison of the mesh using the same settings between the case with a fully resolved well and when the Well boundary condition is used.
This advantage is only useful if we get an accurate solution. Continuing with this test case, we apply a mass flow rate, M_{0}, of 1 kg/s at the well. This corresponds to a mass flux, N_{0}, at the boundary with area, A, of . The mass flux at the edge of length, l, is . The pressure is fixed at the outer infinite element boundaries.
A 1D plot of the pressure along the centerline shows almost perfect agreement with the approaches outside of the well.
Comparison of the pressure along a cut line.
In contrast to specifying the mass flux on the edge, the Well feature accounts for the well radius, even if it is not resolved explicitly. To evaluate the pressure at the well, the Mass Flux edge feature is not suitable, since the expansion in the radial direction is not considered. The Well boundary condition provides a variable, dl.well1.p
, which gives the well pressure.
The Well boundary condition can be used in the geothermal doublet example mentioned earlier. We use a slightly modified version of the model that is featured in the previous blog post. In this case, geothermal groundwater is produced through the production well at a rate of 150 l/s. The water is reinjected at the same rate after it has been used for heat generation with a temperature of 5°C. A horizontal hydraulic gradient of 2 mm/m is applied at the outer boundaries.
Model setup (left) and mesh (right) of the geothermal doublet model.
Prior to COMSOL Multiphysics 5.3, the production and injection wells were drawn as cylinders embedded in the geological formations, and the mass and heat fluxes were defined on the cylinders’ surfaces by using Inlet and Outlet boundary conditions for Darcy’s law and Temperature and Outflow boundary conditions for heat transfer.
Now, the wells are defined by single edges, and the new Well and Line Heat Source features are used for defining the mass and heat fluxes. The settings for both features are shown below.
The settings for the Well (left) and Line Heat Source features (right).
In the Well feature, you specify the mass flow rate at the injection well by setting M_{0} = 150 l/s ρ_{water}. The density of water is specified in the Materials node, which is accessed by the expression mat5.def.rho
. For the line heat source, we define the source term per unit length according to Q_{l} = M_{l}C_{p}ΔT. Here, M_{l} is the mass flow rate per unit length and is automatically calculated by the Well feature (dl.well1.Ml
); C_{p} is the specific heat capacity of water ( mat5.def.Cp
); and ΔT = T_{inj} – T is the temperature difference between the injection temperature and the actual temperature.
With the Well feature, there are about 8% fewer mesh elements (118,000 versus 126,000) as compared to the model that uses cylinders as boreholes; and the simulation runs approximately 10% faster (31 minutes versus 26 minutes). The animation below shows the evolution of the temperature over 5 years.
The evolution of temperature in the geothermal reservoir over five years.
To prove that the Well feature gives the same results as the corresponding boundary condition on a cylindrical surface, we compare the results of the production temperature. We can see that there is very good agreement between the results.
Comparison of the production temperature for both modeling options.
In this blog post, we have seen that the new Well boundary condition can improve performance as well as make it easier to model wells. We have also learned about the background and how the coupling to heat transfer is set up. To see other new features in the Subsurface Flow Module as of COMSOL Multiphysics 5.3, head to the Release Highlights page.
Microgravity is the condition of “free fall” experienced by, for example, objects like satellites that “fall” toward Earth but actually never reach its surface. In this condition, gravity and weight does exist, but is not measurable on a scale. Some people refer to this as zero gravity.
Applying conditions of microgravity to certain systems and processes enables scientists to study them without accounting for effects like hydrostatic pressure and sedimentation. By investigating biological processes exposed to microgravity conditions, we can advance technologies associated with tissue engineering, stem cell research, vaccine development, and more.
One area where microgravity is proving helpful is cancer research. From previous studies, we know that microgravity exposure suppresses immune cell activity and changes genomic and proteomic expressions. As such, scientists are investigating whether these changes also influence cancer development. The goal is to find novel therapeutic targets for metastatic cancer cells by influencing their migration, and therefore their activity.
A research team from SUNY Polytechnic Institute and SpacePharma, Inc. joined forces to develop a culturing system to test how microgravity affects metastatic cancer cell migration. This system isolates gravity as an experimental variable, thereby determining its contribution to cellular function in normal gravity on Earth. Considering the behavior of the culturing system in Earth’s gravity will initially provide insight into how microgravity conditions can be used for lab-scale experiments. The simulation technique will eventually be translated and used for space flight experiments within a low Earth orbit (LEO).
The setup for performing cell culture chips experiments on a chip (left) and a CAD representation (right). Images by A. Dhall, T. Masiello, L. Butt, M. Strohmayer, M. Hemachandra, N. Tokranova, and J. Castracane and taken from their COMSOL Conference 2016 Boston poster.
Running these microgravity experiments in the normal gravity of Earth can be difficult and requires a robust system design. CFD simulation is one way to help understand this problem, augment a good design, and optimize operating and flow conditions.
First, let’s take a closer look at the cell culturing system, which exposes human cancer cells (contained in cell culture chambers) to microgravity conditions. To increase the number of cells during cell maintenance, the system supplies growth media via a media inlet. The system can also reduce the number of cells — and avoid overcrowding — by lifting the cells with trypsin and flushing them out. Another key element in this system is chemoattractants, which influence cell migration.
The initial design of the culturing system. Image by A. Dhall et al. and taken from their COMSOL Conference 2016 Boston poster.
To perform the preliminary analyses of the culturing system, the research team used two interfaces:
When using the Single-Phase Flow interface, the team tested for backflow into the cell culture chamber when the outer channel is flushed with cell growth media. From their results, the researchers found that using either valves or nozzle-diffuser flow can help avoid backflow.
The potential backflow in a cell culture system that occurs due to flushing media through the outer channels of a culture unit. Image by A. Dhall et al. and taken from their COMSOL Conference 2016 Boston presentation.
Simulation was also used to calculate the optimal flow rate range under the chosen operating conditions. In these studies, the researchers modified the culture chip system to contain three chambers, as shown below.
Modified culture chip system with three chambers. Image by A. Dhall et al. and taken from their COMSOL Conference 2016 Boston presentation.
The results, shown below, indicate that when the flow from the inner chamber is less than or equal to the flow from the outer chambers, the cell growth media do not leak into the outer chambers. However, as the flow into the inner chamber increases, the media within the inner chamber spread outward, eventually leaking into the outer chambers via the third channel.
The optimal flow rate range for a three-chamber chip. In these plots, the researchers varied the ratio of the input velocity in the inner chambers (V_{IC}) to input velocity in the outer chambers (V_{OC}) and visualized the resulting flow. Images by A. Dhall et al. and taken from their COMSOL Conference 2016 Boston presentation.
In the image below, the flow in the outer chambers runs opposite to the flow in the inner chamber. The result is that the cell growth media leak through all of the channels. Using the information they learned about the leakage, the team can improve the design of the cell culture chip.
Leakage in the cell culture chip system when the flow of the cell growth media has equal and antiparallel input velocities. Image by A. Dhall et al. and taken from their COMSOL Conference 2016 Boston paper.
The final simulations are of the diffusion of the chemoattractant along a gradient. The chemoattractant has an initial concentration of 0.04 mM and travels through a 0.6-mm migration channel. Simulating this migration shows that the researchers can establish a gradient at a practical timescale for cell migration experiments.
Diffusion of the chemoattractant over time. Image by A. Dhall et al. and taken from their COMSOL Conference 2016 Boston presentation.
Designing a functional culturing system is the key to successfully studying cancer cell migration in microgravity conditions, thus identifying new therapeutic targets for cell mestatatic behavior. In the future, the research team plans to enhance their study by looking into how the cell growth media and chemoattractant interact.
Consider a wing moving in air, say at a constant speed u. The disturbances created by the wing propagate as pressure waves in air (here, sound waves), so these disturbances propagate at the local speed of sound, a. If the speed of the wing is smaller than the speed of sound, u < a, the disturbances propagate faster than the wing itself. Therefore, the fluid upstream is influenced by the presence of the wing before the wing reaches that location. As the speed of the wing is below the speed of sound, this is known as subsonic flow. This phenomenon can also be represented in terms of the Mach number, Ma = u/a < 1.
When the speed of the wing approaches a fraction of the speed of sound — at around Ma = 0.3 — compressible effects become significant. When the speed is greater than the local speed of sound (i.e., supersonic, Ma > 1), the fluid upstream is no longer influenced by the wing before it reaches that location, since the pressure waves have not propagated there yet, as shown in the image below (Mach cone in red). The disturbance propagation gives us qualitative insight into the demarcation between the subsonic and supersonic flow regimes.
The disturbance propagation for subsonic (left), sonic (middle), and supersonic (right) wing speeds, where the circles represent the pressure/sound waves. The arrow represents the direction of movement of the object.
To quantitatively resolve the fluid flow, we first determine whether the flow will be in the laminar flow regime or if it will have transitioned into a turbulent flow regime. The flow regime is based on the Reynolds number, Re = ρuL/μ. Here, ρ, μ, and u are the fluid density, dynamic viscosity, and speed, respectively. The characteristic length (the chord length in the case of a wing) is denoted by L. We then solve either the Navier-Stokes equations with continuity and constitutive relations for laminar flow or the Reynolds-averaged Navier-Stokes equations with continuity and constitutive relations for turbulent flow. Depending on the problem, a suitable turbulence model can be used to resolve the turbulence.
If the flow becomes highly compressible, as is the case for supersonic flows, the energy equation has to be solved in addition to the mass and momentum conservation equations mentioned earlier, whether the flow is in the laminar or turbulent flow regime. The viscous effects in supersonic flows are usually negligible, except in the regions with sharp gradients such as shocks or boundary layers. However, if the viscous effects in the regions of interest are negligible, then removing the viscous terms in the Navier-Stokes equations yields the Euler equations. These inviscid flow equations can be solved analytically for simple geometries.
The supersonic flow field around a diamond airfoil, showing the different regions in the flow field.
For simplicity, consider the benchmark case of supersonic flow past the cross section of a wing with a diamond airfoil. This topic has already been investigated by COMSOL Multiphysics® users in the paper “Numerical Study of Navier-Stokes Equations in Supersonic Flow over a Double Wedge Airfoil using Adaptive Grids“. As mentioned earlier, the fluid upstream (zone 0 in the figure above) is not influenced by the disturbances caused by the airfoil. So when the fluid approaches the airfoil, it faces an abrupt decrease in flow area (similar to a concave corner) and has to suddenly change its direction to match the boundary conditions in zone 1.
This process can only occur if there is a discontinuity in the flow. This discontinuity is known as a shock wave and is very thin; i.e., on the order of the mean free path of the gas molecules, which is around 50 nm in air at atmospheric pressure. When a shock wave is inclined to the flow direction due to the shape of the object and the flow’s Mach number, it is called an oblique shock wave. Across a shock, the static pressure, temperature, and density increase, while the Mach number and total pressure decrease.
As mentioned above, when we assume inviscid flow (which is usually a valid assumption to estimate flow properties in supersonic flows), the shock angle (σ) can be determined for a weak shock based on the inclination of the object. Here, this is the half angle of the diamond airfoil (δ) and the upstream Mach number (Ma_{0}).
For a fluid with specific heat ratio γ, the Mach number after the oblique shock in zone 1 can be estimated from:
After the shock along the diamond airfoil, the flow encounters an expanding area around the top of the airfoil (similar to a convex corner). The change in the flow direction to match the boundary conditions is achieved through an expansion fan, also known as a Prandtl-Meyer expansion fan. Across an expansion fan, the static pressure, temperature, and density decrease, while the Mach number increases. Again, by assuming inviscid flow, we can estimate the Mach number after the expansion fan based on Ma_{1} and the turning angle (θ) from:
Based on the above equations, we can estimate the shock angle and the Mach number across a shock wave or expansion fan. Similar estimates for pressure, density, and temperature can also be computed. However, for more complicated geometries where the shock/expansion waves interact or for highly viscous fluids, it would be more appropriate to resolve the flow through the numerical solution of the conservation equations in their entirety. Let’s model this simple benchmark case of a diamond airfoil and compare the results with estimates from the above equations.
As mentioned earlier, the flow in this scenario becomes highly compressible at supersonic speeds, which leads to a strong coupling of all of the conservation equations. These equations now have to be solved together to resolve the shock waves and the expansion fans that occur in the flow field, which can be done using the High Mach Number Flow interface in the CFD Module. Let’s take a look at how to set up a model of supersonic flow past a diamond airfoil using the High Mach Number Flow interface.
The model configuration, including the boundary conditions.
A schematic of the model setup, along with the static pressure and temperature at the inlet, is shown above. Based on the inlet conditions and airfoil chord length, the Reynolds number is found to be greater than 3 x 10^{5}, indicating that the flow is in the turbulent regime. Therefore, the k-ε turbulence model is used here.
A hybrid flow condition is used under the outlet boundary condition. This condition allows for the flow at that boundary to be either supersonic or subsonic. If the flow is subsonic, then the static pressure specified under the condition’s settings would be ensured. A no-slip condition is defined on the airfoil and a combination of slip wall and outlet (hybrid flow) boundary conditions are used on the top and bottom walls to mimic the inviscid model equations that are solved analytically. In addition, since this is definitely a nonlinear problem, it is important to note that we can achieve better convergence for the problem by specifying initial conditions that could be close to the final solution — as discussed in a previous blog post on nonlinear static finite element problems.
As mentioned earlier, a shock wave is a form of discontinuity in the flow field. Therefore, a very fine mesh must be used at and around the pressure wave in order to numerically resolve this region. The spatial location of the shocks or expansion fans is not known ahead of time. In addition, we might be interested in computing the flow field for varying angles of attack and different inlet conditions, which would result in different locations of shocks in the domain. Therefore, if a uniform mesh is used for such problems, it would need to have a very high mesh density everywhere, significantly increasing the computational cost. However, in the COMSOL Multiphysics® software, we can take advantage of adaptive mesh refinement (an option available in the solvers) to dynamically adjust the mesh refinement in the domain, such that a fine mesh is created near the discontinuities and a comparatively coarser mesh is used in the remaining domain. Here, the conditions for adaptive mesh refinement are defined under the settings for the Stationary study step.
The results from the study and those from the expressions given in the previous section are listed in the table below for comparison. The results from COMSOL Multiphysics are evaluated in the midsection of the zones along the streamline, shown in the streamline plot below. The simulation results and theoretical estimates are within 3% of each other, as indicated by the tabulated results. It is also important to note that any viscous effects present in the problem are taken into account by the numerical model, making it more physically accurate than the inviscid theory on which the analytical estimates are based.
Zone Number | Mach Number (Theory) | Mach Number (Model) | Static Pressure (atm) (Theory) | Static Pressure (atm) (Model) | Static Temperature (K) (Theory) | Static Temperature (K) (Model) |
---|---|---|---|---|---|---|
0 | 2.0 | 2.0 | 1.0 | 1.0 | 300 | 300 |
1 | 1.45 | 1.44 | 2.22 | 2.21 | 378 | 381 |
2 | 2.55 | 2.53 | 0.40 | 0.41 | 231 | 237 |
A contour plot of the static pressure (in Pa) along with the streamlines (in blue) for zero angle of attack, an inlet Mach number of 2.0, pressure of 1 atm, and temperature of 300 K.
A comparison of the shock angle is qualitatively shown in the Mach number surface plot below. At the location of the shock, a sudden change in the Mach number is observed. In the following plot, a black line has been added to indicate where the shock would be located based on the theoretical estimates. There is good agreement between the theoretical estimates and the simulation results.
A surface plot of the Mach number. The black lines indicate the location of the shock based on expressions for shock angle, zero angle of attack, an inlet Mach number of 2.0, pressure of 1 atm, and temperature of 300 K.
Below, we can see the temperature surface plots for varying angles of attack. In addition to the temperature changes on the top and bottom sides of the airfoil, the change in the shock angle with the change in angle of attack can be clearly observed in these plots. At a 10° angle of attack, the angle of inclination at the bottom becomes 25° (as the half angle is 15°). The analytical theory predicts detachment of the shock, which is also observed in the numerical analysis (shown in the figure below on the right).
A surface plot of the temperature (in K) for varying angles of attack (α) for an inlet Mach number of 2.0, pressure of 1 atm, and temperature of 300 K.
In this blog post, we have discussed supersonic flow characteristics such as shocks and expansion fans. With an example of supersonic flow over a diamond airfoil, we have shown how these characteristics can be resolved in COMSOL Multiphysics using the High Mach Number Flow interface. The shock angle and flow properties on the airfoil from our simulations are in good agreement with analytical estimates from inviscid compressible flow theory.
For more information on modeling supersonic or transonic flows, check out these example models from the Application Gallery:
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Vacuum technology is found in many high-tech applications, including semiconductor processing, mass spectrometry, and materials processing. This technology creates low-pressure environments by using vacuum pumps to remove air molecules from enclosed vacuum chambers.
One type of vacuum pump is a turbomolecular pump, which consists of a bladed molecular turbine. The blades of modern turbomolecular pumps rotate extremely quickly, reaching speeds as high as 90,000 rpm.
A turbomolecular pump.
The momentum transfer from the rotating blades to the gas molecules compresses the gas, which is moved from the inlet to the outlet by the blades. As a result, the pump is able to generate and maintain a high vacuum on the inlet side of the blades. This pumping process is more efficient in the free molecular flow range, since the gas particles mostly collide with the rotor and not with each other.
We can better understand and design turbomolecular pumps by modeling them with COMSOL Multiphysics. But first, we need to figure out the best way to do so.
Instead of focusing on the whole turbomolecular pump, our model geometry depicts part of a single turbomolecular pump stage (a row of blades). Using the model, we calculate gas molecule trajectories in the empty space between the blades. This enables us to assume sector symmetry in the modeling domain.
Model geometry of one sector of one stage of a turbomolecular pump. Gray represents the space between two blades, green represents the blade walls, and black represents the rotor root.
While we don’t use it here, one way to solve the model equations and calculate the pump’s performance in a free molecular flow regime is with the Free Molecular Flow interface from the Molecular Flow Module. This interface is an efficient option and is useful in cases where the molecules of extremely rarefied gases move significantly faster than any object in the modeling domain. However, in turbomolecular pumps, the speed of the gas molecules is comparable with the blade speed. As such, we need a different approach for this problem.
The turbomolecular pump example model.
We use a Monte Carlo approach and the Rotating Frame feature (new to the Particle Tracing Module in version 5.3 of COMSOL Multiphysics®) to automatically apply the fictitious Coriolis and centrifugal forces to the particles. This enables us to compute the particle trajectories within a noninertial frame of reference that moves along with the blades.
This method provides accurate results on how the blade velocity ratio affects the pumping characteristics, such as the maximum compression ratio, transmission probability, and maximum speed factor. We base these characteristics on the transmission probability of argon atoms from the inlet to the outlet and vice versa.
For more information on how we created this model, including the geometric parameters and assumptions, check out the documentation for the turbomolecular pump tutorial.
Let’s begin by computing the transmission probabilities for particles propagating in the forward (inlet to outlet) and reverse (outlet to inlet) directions. As expected, when the blades are at rest, these probabilities are about equal. This is because there is no distinction between the two directions.
However, when the rotation of the blades begins to increase, the particles are more likely to be transported forward through the pump, as the walls successfully transfer momentum to the argon atoms. This corresponds to an increasing compression ratio.
The fraction of particles transmitted in the forward direction (left) and the reverse direction (right) as a function of blade velocity ratio.
We also investigate how the compression ratio and speed factor are affected by the blade velocity ratio. To produce enough compression and speed, pumps use multiple bladed structures comprised of several disks and different types of blades. Blades close to the inlet have a high pumping speed and low compression ratio, while blades close to the outlet have the opposite characteristics.
When the velocity of these blades increases, as seen in the plots below, the maximum compression and speed factor increase. This confirms that the two blade types work together to enhance the performance of the pump.
The effect of blade velocity on the maximum compression ratio (left) and maximum speed factor (right).
This example highlights the new modeling features that enable you to more easily analyze turbomolecular pumps. Try it yourself by clicking on the button below.
A frequent topic on the COMSOL Blog, heat sinks dissipate excess heat in electronic devices. Heat sinks have become more important with advancing technology. As electronic devices become more powerful, they often generate more heat. This can impair the performance of these devices and even reduce their lifespans.
A heat sink on a motherboard. Image by Adikos — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
To analyze the cooling capacity of a heat sink, we can turn to the heat transfer modeling capabilities of COMSOL Multiphysics. Due to the flexibility of the software, we can approach the analysis in multiple ways. Let’s discuss two of these approaches using the simple example of electronic chip cooling.
For our example, we model an electronic chip with a heat sink that is cooled by the surrounding fluid (air). To create the model, we use the heat sink geometry parts available in the Part Library in the Heat Transfer Module, an add-on product to COMSOL Multiphysics. These parts, which are available as of version 5.3, make it easier to include a heat sink in a model geometry.
For more details, check out the Heat Transfer Module Release Highlights page.
In this case, the model consists of an electronic chip and aluminum heat sink. The heat sink is mounted within a channel that has a rectangular cross section. As for the electronic component, it dissipates the equivalent of 5 W of heat. This is distributed throughout the chip volume.
Model geometry of the heat sink (gray) and electronic chip (purple).
Air flows from the inlet (where the temperature is set) through the channel to the outlet. There, convection is the main form of thermal energy transport. A combination of conduction and convection transports the thermal energy in the cooling air, while conduction transfers thermal energy in the electronic component and heat sink.
The model enables us to solve for the thermal balance between the various components and to find the thermal contact between the chip and the heat sink when dealing with a thermally thick layer. We can estimate the cooling capacity of the heat sink and predict the temperature of the electronic component by solving the model equations. Next, let’s look at the two different approaches for solving this model, starting with the more computationally efficient approach.
For the first approach, we only model solid parts and use a convective cooling boundary condition on the heat sink boundaries instead of computing the flow velocity, pressure, and temperature of the air channel. This enables us to make rapid computations. However, the accuracy of these computations is tied to the reliability of the heat transfer coefficient used to define the convective cooling boundary condition. Here, we use an empirical value of 10 W/(m^{2}·K).
Using this approach, we test three configurations for thermal contact between the heat sink and chip:
Temperature plots for the ideal contact (left), air layer (middle), and thermal grease layer (right) configurations.
As expected, the lowest maximum temperature (around 84°C) is obtained for the ideal contact case. When we include an air layer between the heat sink and chip in the model, this rises to almost 95°C, causing the performance of the cooling system to decrease.
We can reduce the effect of this thermal resistance by swapping out the layer of air for a layer of thermal grease. The thermal grease layer improves the thermal contact between the base of the heat sink and the top of the electronic chip. The simulation predicts that the maximum temperature nears that of the ideal thermal contact case. From this, we can conclude that a thin layer of thermal grease helps to improve the heat sink design.
Moving on, let’s take a more computationally expensive approach that adds a domain representing an air channel to the model. This domain allows us to calculate the temperature and flow fields of the air while assuming nonisothermal flow in the channel. This approach is more general than the previous approach, which is helpful, since the heat transfer coefficient is generally unknown. This way, we can accurately simulate flow cooling without needing to approximate the heat transfer coefficient.
The results from this approach show that the maximum temperature of the electronic component is around 95°C. In addition, there is a hot wake behind the heat sink, indicating its convective cooling effects. This second modeling approach is also more precise. For example, we can visualize and predict the temperature difference on a fin at the edge facing the flow or at the opposite side.
The temperature field on the heat sink’s surface and channel walls (left). Visualizing the temperature difference on a fin (right).
We can enhance this approach by modifying the model to account for a large (close to 1) surface emissivity. Since surface-to-surface radiation should be considered when working with such large emissivities, we include it in our simulation. This addition lowers the maximum temperature to around 81°C, as seen below. This reaffirms that when the surface emissivity is large, radiative heat transfer should be considered.
The temperature field when surface-to-surface radiation is included.
As we’ve seen here, COMSOL Multiphysics offers tools that make it possible to use different approaches to analyze electronic cooling — an important step in the design of heat sinks for optimized electronic devices. Each modeling approach has its own benefits, whether it be a more computationally effective option or a method with higher accuracy.
When molten metal solidifies, grains begin to form. These affect the physical properties of the solid metal; for instance, a smaller grain size makes the metal stronger. The grain of a metal is affected by many factors, such as temperature and cooling time. Engineers can also adjust grain morphology during the metal solidification process by using the physical phenomenon of acoustic streaming (AS) to induce drag on particles.
Molten metal being processed. Image by Goodwin Steel Castings — Own work. Licensed under CC BY-SA 2.0, via Flickr Creative Commons.
In AS, an oscillating sonotrode sound emitter is placed in a liquid and generates a steady fluid motion. To produce a significant effect, the sound waves need a high amplitude and frequency, often in the ultrasound range. Thus, this technique can require an ultrasonic treatment.
To further improve and develop AS, engineers rely on costly physical experiments. Simulation provides an alternative, enabling those in the metal processing industry to build models to test AS treatments that use different materials and fluids. These models can then be validated via experimental testing.
Let’s take a look at how researchers from the Institute of Thermal and Fluid Engineering and Institute of Product and Production Engineering at the University of Applied Sciences Northwestern Switzerland investigated this possibility.
The research team’s goal was to build and experimentally validate an AS model that can analyze a variety of fluids by altering the parameters. The resulting 2D axisymmetric model represents an experimental setup of an oscillating sonotrode placed into a fluid and generating a harmonic acoustic pressure field. They simplified this model by assuming isothermal behavior, neglecting cavitation, and using an averaged stationary flow.
Acoustic streaming sample geometry. The numbered dots represent the boundary positions and the colored dots show the locations of the three tracing massless particles. Image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
Since AS is a multiphysics phenomenon, the researchers accounted for two physics phenomena:
Note that accounting for the acoustics in the high-frequency domain requires a compressible fluid description, due to the coupling of density and pressure perturbations.
The researchers solved the model equations with three study steps:
The first study shows that the sonotrode acceleration causes a sharp rise in acoustic particle velocity. Using the acoustic velocity field found here, the force term for the second study is generated.
Frequency-domain results, showing the acoustic velocity field. Image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
The fluid flow study investigates the flow pattern, beginning with the axial jet exiting from the actuating sonotrode tip. The jet continues to the bottom wall, where it is deflected and creates vortices in the bottom corners. The generated flow is almost at a standstill in the open interface zones and has the highest value beneath the sonotrode.
Left: Stationary velocity field for an aluminum melt with a frequency of 20 kHz and 30 µm amplitude. Right: Comparison of the velocities of the three massless particles, with the colors matching those in the sample geometry. Images by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
When looking at the dispersal of the three separate tracing particles, the model demonstrates that the particles beneath the sonotrode (depicted as the black line in the right plot above) have a high acceleration, which increases the number of cycles.
To experimentally validate their simulations, the researchers created a small-scale laboratory model involving an aluminum sonotrode partially submerged in a fluid-filled crucible. Next, they performed experiments at a frequency of 20 kHz and at three different amplitudes: 10, 20, and 30 µm. To track the fluorescent particles used in the experiment, the team relied on a combination of a high-speed camera, diode laser, and lasersheet. They then determined the correlated velocity field via particle image velocimetry.
The experimental setup. Original image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
When testing a seed oil liquid, we can see that the axial jet is visible in both the simulation and experimental results, as shown below. While the results don’t completely line up, they do match on the right side of the crucible for the direction and location of the induced flow.
The velocity fields for the simulation (left) and seed oil test case (right) with a frequency of 20 kHz and amplitude of 30 µm. Images by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
We can also compare the velocities along the rotational axis in the simulation and experiment, which show good agreement near the tip of the sonotrode. The results begin to differ as we move farther away from the tip, with the simulation achieving a peak velocity (over twice that of the experimental maximum velocity) within 10 mm from the tip. The difference between the results decreases when the axial difference increases, with both the simulation and experiment showing a decline behavior.
Comparison of the simulation and experimental results for the velocity magnitude along the rotational axis. Image by D. Rubinetti, D. Weiss, J. Muller, and A. Wahlen and taken from their COMSOL Conference 2016 Munich paper.
The deviation between the results may be due to a few reasons, including nonaccurate optical measurements (the experimental data is difficult to collect) and the team’s simplified simulations. The underlying issue may be that the flow is not steady in reality. As their experimental plot above confirms, momentum dissipates a lot more in the experiments than the model. This suggests that there are unsteady smaller eddies that transfer momentum at a rate that is not described by the averaged steady flow used in the model.
Through their experimental testing, the research team concluded that their AS model gives a qualitative description of the flow, except for the small region close to the sonotrode where the description seems to be fairly accurate. As a reasonable approach for analyzing AS and predicting fluid flow behavior, it can save time and money by minimizing the required amount of physical experiments.
Simulation is also a good choice for testing various fluids, parameters, and geometries. This enables engineers to efficiently study different AS treatments by tailoring the model to fit a specific case. The researchers also note that the model is versatile enough to be used to study other sound-driven fluid motion applications.
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.”
Probe tubes are attached to microphone cases in order to distance the device from the sound field being measured. When fitting hearing aids, the tube is inserted into the ear canal with the microphone worn on the outside of the ear. This system provides measurements that calibrate and verify the comfort and effectiveness of hearing aids, specifically if the device amplifies signals to the level that the patient needs. In fact, the American Speech-Language-Hearing Association and the American Academy of Audiology say that in-the-ear measurements are the preferred method for verifying hearing aid performance.
A probe tube microphone performing in-the-ear measurements. Image by Cstokesrees — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
When adding a probe tube to a microphone, we have to consider how these two components interact with one another. For instance, we need to understand how the probe affects the sensitivity of the microphone, thus the measurements that the device delivers. As we show here, multiphysics simulation provides answers.
For this example, we use a time-dependent model that consists of a generic probe tube microphone configuration. It includes:
The probe tube is made of an elastic material with a Young’s modulus of 0.1 and a Poisson’s ratio of 0.4. In the schematic below, L represents the length of the tube, while D_{0} is its outer diameter. The cavity in front of the microphone is a cylinder with a radius of R and height of H. This cavity is connected to a cone that has a bottom radius of R and top radius of D_{0}. An external sound field with the wave vector k hits the probe tube. Note that this sinusoidal wave moves in the positive x direction and has an amplitude of 1 Pa.
The probe tube microphone configuration.
To model this probe tube microphone design, we use the Pipe Acoustics, Transient interface. In our analysis, the probe tube is treated as a 1D structure — a valid assumption as long as we neglect the interaction between this component and the incoming sound field. We also assume that no significant thermal and viscous boundary losses occur inside the tube. This holds for the current configuration in which the incident field is a monochromatic wave. Since the diaphragm is not a fully rigid structure, we assume a resistive loss that is consistent with the impedance of common condenser microphones. This gives us a fully coupled acoustics simulation, as the probe tube is connected to two separate 3D pressure acoustics domains.
When analyzing a probe tube microphone, an important parameter to consider is the relationship between the pressure at the tip of the probe and the pressure at the diaphragm. This is a necessary step for calibrating the measurement system. The plot below on the left shows that the solution, following an initial transient, becomes periodic after around 4 ms. The system then experiences a gain of about 1.4 and a phase shift takes place. These two factors are dependent on the frequency of the applied signal, which is a pure harmonic tone of 500 Hz. The plot on the right depicts the pressure distribution in the xz-plane at the end of the time interval.
Left: Diaphragm pressure vs. probe tip pressure. Right: Pressure distribution in the xz-plane at 8 ms.
These results show the potential of using the COMSOL Multiphysics® software to analyze probe tube microphone designs. With a better understanding of how the probe tube and microphone interact, it is possible to further improve the design of these systems for fitting hearing aids and for other applications.
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 model using low Reynolds number 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.