Question: what is the difference between “incompressible” and “constant density?”

  • Compressible flows are high speed relative to the sound speed.

  • When changes in fluid velocity are high enough to significantly impact fluid properties, we say a flow is compressible.

  • From the Bernoulli equation we have $$\Delta P = \frac{\rho \Delta v^2}{2}$$ Using $c^2 = \gamma P/\rho$, and letting $\Delta v = v$, we have $$\frac{\Delta P}{P} = \frac{\gamma}{2}Ma^2,$$ where $\gamma = c_p/c_v$.

  • For v=100 m/s, $\Delta P/P_{atm}$ is 6%.

    • 100 m/s is 233 mph and gives $Ma=0.288$ for air at 1 atm.
  • For v=347 m/s, $\Delta P/P_{atm}$ is 70%.

    • v=347 m/s is 776 mph and gives $Ma=1$ for air at 1 atm.

We can have a variable density flow that is considered incompressible.

  • Density variations may occur due to, e.g. temperature or species composition changes.
  • Incompressible for $Ma \le 0.3$ (approximately).

Low Ma flows have pressure fluctuations due to the flow that do not significantly affect the flow.

Flow solvers: stable stepsize

Unfortunately, stable explicit numerical solvers have step sizes set by the smallest timescales, even if the corresponding physical process doesn’t significantly affect the flow.

  • We would like our stepsize to be set by an advective CFL proportional to $\Delta x/u$.
  • For stability, a compressible solver requires an acoustic CFL proportional to $\Delta x/(u+c)$.
  • If Ma = u/c is small, then we need many fewer timesteps for stability than we need for accuratly representing the flow. This defines a stiff problem.
  • The number of stepsizes needed for stability relative to the number needed for accuracy is $(\Delta x/u)/(\Delta x/(u+c)) = 1 + 1/Ma$.

Pressure gradient scaling

  • For the Euler equations , the wave speeds are $u$, $u-c$, and $u+c$.
  • In following the derivation of these, $c$ can be replaced with $\sqrt{\beta}c$, where $\beta$ is a constant, if we multiply the pressure gradient in the momentum equation by $\beta$.
    • This is equivalent to adjusting the gas constant $R$.
    • The gasdyn code illustrates this.
      • The stable timestep size varies approximately as $\sqrt{\beta}$. In the code, $\Delta t\approx 0.088,\mu s$ for $\beta=1$, and $\Delta t\approx 1.2,\mu s$ for $\beta=0.005$. This timestep size ratio is 0.0733, which is practically the same as $\sqrt{0.005}=0.0707$.
  • In this case the relative number of timesteps required compared to that for advection only is $1 + \sqrt{\beta}/Ma$.
    • The relative change in pressure is $\gamma Ma^2/(2\beta)$.
  • If Ma = 0.1, $\beta = 1.0$ the relative steps are 11 and $\Delta P/P = 0.7%$.
  • If Ma = 0.1, $\beta = 0.1$ the relative steps are 4 and $\Delta P/P = 7%$.
  • Hence, scaling $\beta$ has allowed fewer timesteps, but has artificially increased pressure fluctuations. The pressure fluctuations increase faster than the timesteps decrease.

Pressure Projections

  • Low Mach flows can also be solved using pressure projection approaches.
  • References:
    • “Computational Methods for Fluid Dynamics,” J. H. Ferziger, M. Peric, 3rd edition, Springer, 2002
    • “Numerical Heat Transfer and Fluid Flow”, S. V. Patankar, Hemisphere Pub., 1980.
    • “Numerical Heat Transfer and Fluid Flow”, S. V. Patankar, Hemisphere Pub., 1980.
    • Saad et al. (2018)

Constant density, unsteady

  • Following Ferziger

We have the continuity and momentum equations, which are two equations in pressure and velocity:

$$\nabla\cdot v = 0,$$ $$\frac{\partial v}{\partial t} = -\frac{\partial v v}{\partial x} - \nabla\cdot\tau - \frac{1}{\rho}\nabla P.$$

  • The momentum equation is an equation for velocity.
  • There is no explicit pressure equation.
  • The continuity equation provides a constraint on velocity.
  • The pressure is a scalar field that will be set so that the continuity equation is satisfied. (Note that the pressure field is only specified up to an additive constant, since only $\nabla P$ appears.)

Apply a simple Explicit Euler timestep to momentum:

$$v^{n+1} = \underbrace{v^n + \Delta t\left(-\frac{\partial v v}{\partial x} - \nabla\cdot \tau \right)}_{H^n} - \frac{\Delta t}{\rho}\nabla P.$$

Take the divergence of this (which is zero), and rearrange to get the pressure equation:

$$\nabla^2P = \frac{\rho}{\Delta t}\nabla\cdot H^n.$$

To advance a given step, we solve this equation for P. And then use this in the momentum equation for $v$, which will satisfy continuity by construction.

$$v^{n+1} = H^n - \frac{\Delta t}{\rho}\nabla P.$$

Variable density, unsteady

  • Saad et al. (2018) extend the treatment above to the case of variable density flows, where the key difference is the treatment of the nonzero velocity divergence.

Steady treatments

  • For steay flows (as in RANS), variations of the SIMPLE algorithm are common. See Patankar and Ferziger.