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

Given

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}:

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}:

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:

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 forces extremely small time steps to be used for explicit time marching even when one is not interested in disturbances traveling at acoustic speeds. This makes the convergence to steady state extremely slow. Moreover, in the limit of vanishing Mach number, discretization based on density-based approach introduces error that grows inversely with Mach number. This is true regardless of whether one uses explicit or implicit time marching.

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.

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 evolved. Current methods based on state-of-art density-based approach can be applied to low Mach number flows with compressible eos as well as flows involving constant-density equation of state.

**Goal**Given

- the solution at t=0 along with
- appropriate boundary conditions at all physical boundaries (at all times),

**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

**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

**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.

**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 forces extremely small time steps to be used for explicit time marching even when one is not interested in disturbances traveling at acoustic speeds. This makes the convergence to steady state extremely slow. Moreover, in the limit of vanishing Mach number, discretization based on density-based approach introduces error that grows inversely with Mach number. This is true regardless of whether one uses explicit or implicit time marching.

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 evolved. Current methods based on state-of-art density-based approach can be applied to low Mach number flows with compressible eos as well as flows involving constant-density equation of state.

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