Numerische

The Propagation of Information in Subsonic Fluid Flows

I have often heard engineers working in the industry comment on some property of the Navier-Stokes equations. Often times when a puzzling CFD result is being reviewed by a team of engineers and something unexplainable has happened one engineer in the room will make a comment like “Well, subsonic flows are elliptic which means that information can propagate upstream so maybe our geometry change influenced the rest of the domain.”

We will go over the phrase “subsonic flows are elliptic”, where people likely learned this concept from, and review how hyperbolic and elliptic partial differential equations can propagate information in multiple directions.

Background on Classification of Partial Differential Equations

Partial differential equations (PDE) play an important role in mathematical physics and fluid dynamics. Most introductory books on mathematical physics cover the development of canonical PDEs associated with physical phenomena. Here are three of the most common and useful examples when discussing fundamental transport phenomena such as transient heat conduction, wave propagation in continua and steady-state diffusion processes.

These PDEs can help us understand the behavior of much more complex systems of PDEs that are used in Computational Fluid Dynamics (CFD) and, more generally, Computational Science. There are many misconceptions about the classification of systems of equations such as the Navier-Stokes. Many people will adopt statements from very specific mathematical developments found in senior undergraduate texts and try to apply those learnings back to the Navier-Stokes equations. Similarly, there are misconceptions about the Euler equations which are much easier to classify than the Navier Stokes which contains added complexity of viscous and heat conduction terms.

In this post a more fundamental and phenomenological view of the Euler and Navier-Stokes equations is taken to help the reader understand the behavior of different terms in each of these complex systems. For the sake of further exploration, there are many texts which cover a majority of these developments in great detail. Randal Leveque’s book Finite Volume Methods for Hyperbolic Problems1 is one of the most complete and consistent treatments of numerical solution techniques for hyperbolic problems. This class of PDEs is very important in the field of CFD due to the presence of convective transport terms in both the Euler and Navier-Stokes equations.

Canonical PDEs and Their Classification

One dimensional examples are a useful and practical starting point for discussions regarding the classification of PDEs and how it informs their numerical solution. The solution to a 1D PDE will typically be denoted as u(x, t) where x is the the spatial coordinate and t is time. In general, x\in\mathbb{R} and t\in\mathbb{R^+}. For a specific problem on interest, boundary conditions will be applied where the value of u or its derivative will be specified.

The Heat Equation

\frac{\partial u}{\partial t} - \alpha \frac{\partial^2 u}{\partial x^2}=0
u=f(x,t) \text{\; where \;} x\in\mathbb{R} \text{\; and \;} t\in \mathbb{R}^+ \text{\; and \;} \alpha=\frac{k}{\rho c_p}

The Wave Equation

\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2}=0
u=f(x,t) \text{\; where \;} x\in\mathbb{R} \text{\; and \;} t\in \mathbb{R}^+

Laplace’s Equation

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
u=f(x, y) \text{\; where \;} x\in\mathbb{R} \text{\; and \;} y\in \mathbb{R}^+

The analytical techniques used to solve these equations are covered in many texts on mathematical physics. Numerical methods for solving such equations require an understanding of their mathematical character via classification. The canonical PDEs shown above are fairly straightforward to classify using the following equation for a general second order linear partial differential equation.

A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial t} + C\frac{\partial^2 u}{\partial t^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial t} + Fu=G

The second variable doesn’t necessarily have to be t. For instance, it can be y as found in Laplace’s equation:

A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + Fu=G

In order to classify one of our canonical PDEs we can use the general forms shown above and evaluate its discriminant i.e. B^2-4AC.

  • B^2-4AC > 0 \implies PDE is hyperbolic
  • B^2-4AC = 0 \implies PDE is parabolic
  • B^2-4AC < 0 \implies PDE is elliptic

In the case of the heat equation:

B=0, A=\alpha, C=0
or
B^2-4AC=0

In the case of the wave equation:

B=0, A=c^2, C=1
or
B^2-4AC=0-(-c^2)=c^2>0

In the case of Laplace’s equation:

B=0, A=1, C=1
or
B^2-4AC=0-(1)(1)=-1<0

Overall, the heat equation is parabolic, the wave equation is hyperbolic and Laplace’s equation is elliptic. Now that we have that background let’s look at how some modern aerodynamic texts use the approach shown above to classify PDEs that govern fluid flows.

The Case of Linearized Velocity Potential Equation

In fundamental studies of aerodynamics it is convenient to reduce the Navier-Stokes equations to a more tractable equation called the Linearized Velocity Potential Equation (LVPE).

(1-M_{\infty}^2)\frac{\partial^2 \hat{\phi}}{\partial x^2} + \frac{\partial^2 \hat{\phi}}{\partial y^2} = 0

This equation and its background can be found in Anderson2. Please take note that Mach number must be be greater than 0. Since the linearized velocity potential equation is linear and second order like the equations we classified above we can use the same approach to classify it.

B=0, A=(1-M_{\infty}^2), C=1
or
B^2-4AC=0-(1-M_{\infty}^2)(1)=M_{\infty}^2-1

Instead of a simple classification like the ones we encountered for canonical PDE this one shows something dependent upon Mach number.

If M_{\infty}^2 -1 <0 then the free stream Mach number must be less than 1. So, for a subsonic flow (M_{\infty}<1) the linearized velocity potential equation is elliptic.

Now, if M_{\infty}^2 -1 >0 then the free stream Mach number must be greater than 1. So, for a supersonic flow (M_{\infty}>1) the linearized velocity potential equation is hyperbolic.

It is at this point that most readers will use the information above and extend it to many different fluid flows. This is not always appropriate and warrants further discussion.

The assumptions made in reducing the Navier-Stokes equations to the LVPE are:

  • Irrotational
    • \omega=\nabla \times u = 0 where \omega is referred to as vorticity
  • Inviscid
    • \mu = 0 where \mu is the dynamic viscosity of the fluid of interest
  • Isentropic
    • Entropy is constant throughout the flow field. This assumption neglects the effects of irreversible processes such as heat transfer through walls, fluid friction, losses through shock waves, etc. Mathematically, it means, S_2 - S_1 = nC_p \frac{T_2}{T_1} - nR\frac{p_2}{p_1} = 0

For external flows like the freestream conditions found in aerodynamics of aircraft configurations and airfoils (in isolation) these types of assumptions are reasonable. For internal flows found in propulsion systems these assumptions can lead to significant errors.

When these restrictions are alleviated one at a time the full Navier-Stokes equations will eventually be recovered. However, if one relaxes the irrotational assumption listed above the compressible Euler equations will be recovered.

The compressible Euler equations are a hyperbolic system of PDEs which describe fluid flows which are inviscid and isentropic. The Euler equations are hyperbolic regardless of the Mach number of the flow. Overall, they behave like

  1. LeVeque, Randall J. Finite Volume Methods for Hyperbolic Problems. of Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press, 2002. ↩︎
  2. Anderson, John D. Fundamentals of Aerodynamics. 7th ed. New York: McGraw-Hill Education, 2024. ↩︎

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