Thursday, October 24, 2019

Density-Based Approach vs Pressure-Based Approach

Let us consider unsteady Euler equations along with a compressible equation of state (eos) - for example ideal gas.

Goal

Given
  • the solution at t=0 along with
  • appropriate boundary conditions at all physical boundaries (at all times), 
calculate the "time-accurate" solution for t>0.

Time Marching

Each transport equation (continuity + momentum + energy) has a time derivative. So the simplest approach is to march each transport equation in time - explicitly or implicitly. That gives us {rho, rho*u, rho*v, rho*w, rho*E} at next time level. {rho, rho*u, rho*v, rho*w, rho*E} can be translated to {rho, u, v, w, T}:
  • {rho, rho*u, rho*v, rho*w, rho*E} => {rho, u, v, w, E}
  • {rho, u, v, w, E} => {rho, u, v, w, e} using E = e + (u*u + v*v + w*w)/2
  • {rho, u, v, w, e} => {rho, u, v, w, T} using e = Cv*T
So we have solved for 5 primary variables by marching the transport equations in time. What about the 6th primary variable (p)? p is obtained using the eos. This numerical approach is often termed as density-based approach because density is "solved" for by time marching the continuity equation while pressure is "updated" using eos.

What if we have a Constant Density EOS?

With constant density eos, the time derivative term in continuity equation disappears. What can we do now? We still have time derivatives in momentum and energy equations. We can march these equations in time to get {u, v, w, E} at next time level. {u, v, w, E} can be translated to {u, v, w, T}:
  •  {u, v, w, E} => {u, v, w, e} using E = e + (u*u + v*v + w*w)/2
  •  {u, v, w, e} => {u, v, w, T} using e = Cp*T
Now we have solved for 4 primary variables by marching the transport equations in time. rho is already known but we still need to compute p. Unfortunately the eos (rho = constant) does not help here in updating p! But there is one transport equation that we haven't utilized yet - the continuity equation. Although there is no pressure term in continuity equation, it can be applied to momentum equation to obtain a Poisson equation for pressure. This pressure Poisson equation can be used to update p. This numerical approach is called pressure-based approach because pressure is the primary variable (as opposed to density) which is "solved" for using a transport equation. There, however, is an important thing to note here. When momentum equations are marched in time to compute velocity they have to use the old pressure. This velocity satisifies momentum equation but does not satisfy continuity equation. After this the pressure Poisson equation (which is representative of continuity equation) is solved to update pressure. So the momentum equations and pressure Poisson equation need to be solved iteratively several times one after another at each time level.

Let us redefine Density-Based Approach

We said that density-based approach is called so because we solve for density (and not pressure) using the continuity transport equation. But that is not entirely true in modern perspective. Today, a method can be said to follow density-based approach if it has following features:
  • all the transport equations (continuity + momentum + energy) are marched in time (or pseudo-time if only interested in steady state solution) either explicitly or implicitly, and
  • convective fluxes are computed using a Riemann solver based on the eigenstructure of all the transport equations (continuity + momentum + energy) taken together.
Note that this new definition does not put any emphasis on solving for density using a transport equation and updating pressure using eos. Even if a method solves for pressure using a transport equation and updates density using eos it can be classified as density-based approach if it fulfills above criteria.

Low Mach Number Flows

Let us get back to compressible ideal gas eos. As mentioned above, a density-based approach needs to compute convective fluxes using a Riemann solver based on the eigenstructure of all the transport equations (continuity + momentum + energy) taken together. Even for multi-dimensional problems, usually a 1D Riemann solver is used. The eigenvalues of the system used in 1D Riemann solver are {un, un, un, un + c, un - c} where un is the face-normal velocity and c is the speed of sound. At low Mach number, un << max{|un + c|, |un - c|}. This huge disparity in eigenvalues poses problems with both explicit and implicit time marching:
  • for explicit time marching time step size needs to be extremely small, and
  • for implicit time marching the linear system becomes extremely stiff.
Although the pressure-based approach was originally designed for constant-density equation of state, over the years it has been modified and adapted to be used for compressible equations of state as well. Because pressure-based approach does not use convective fluxes based on eigenstructure of all the transport equations (continuity + momentum + energy) taken together, it does not face the same problems as density-based approach. Because of this, pressure-based approach is widely used to solve low Mach number flows for compressible equations of state.

Current State

As mentioned above, pressure-based approach has been adapted over years to be to be used for compressible equations of state as well. The state-of-art methods based on pressure-based approach can be applied to the whole range of flows involving compressible equations of state, not just limited to low Mach number flows.

The same can be said about state-of-art methods based density-based approach. Over the years, density-based approach has also been updated to be applied to low Mach number flows with compressible eos. Current methods based on density-based apporach can also be applied to flows involving constant-density equation of state.

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