Type of PDE: Hyperbolic or Parabolic or Elliptic?
 

When reading the book, "Computational Fluid Mechanics and Heat Transfer", 2nd adition, by Tannehill, Anderson and Pletcher, on page 624, I read the following: "the unsteady compressible N-S equations are a mixed set of hyperbolic-parabolic equations in time.If the unsteady terms are dropped from these equations, the resulting equations become a mixed set of hyperbolic-elliptic equations, ..." My question is: why are those equations a mixed set of hyperbolic-parabolic or hyperbolic-elliptic?

Thank you!

Re: Type of PDE: Hyperbolic or Parabolic or Ellipt
 

(1). I think, it has something to do with the local flow behavior. (2). If the local flow behavior is something like transient heat conduction, then it is parabolic. (3). If the flow behavior is subsonic, steady state, then it is elliptic. (4). If the flow behavior is supersonic, steady state, then it is hyperbolic. (5). If the flow behavior is transient inviscid, then it is hyperbloic. (6). In other words, flow field solution can exhibit different flow behaviors in different part of the flow field. (thus the word "mixed" is used) (7). There are also standard method to classify the type of a partial differential equations, which I think, is also included in the Anderson' book "Computational Fluid Mechanics and Heat Transfer". (8). The important thing is to understand the local flow behavior based on the local Mach number, Reynolds number, and its transient state. This is important because you can run into a steady-state supersonic flow (hyperbolic)with embedded subsonic pocket or region (elliptic). In that case,solution method and boundary conditions must be changed according to it type. I think, you are in the PhD. domain now.

Re: Type of PDE: Hyperbolic or Parabolic or Ellipt
 

Re: Type of PDE: Hyperbolic or Parabolic or Ellipt
 

Thank you for the answer. I understand the basis of the classification from your words. But it seems to me that, from your answers, the judgement of the types of the N-S equations to be hyperbolic-parabolic or hyperbolic-elliptic is based on the physical characteristics of the solutions which are not known before we get them or if we don't have experience in heat transfer or fluid mechanics. So, like other judgement for rather simple 2nd order PDE's by using b^2-4ac to decide the type which can be carried out by people who have only mathematical knowledge, are there any mathematical methods which can be applied directly to the N-S equations to decide the mixed types? Or, the types can only be judged based on the computational outcome or on some experience?

Re: Type of PDE: Hyperbolic or Parabolic or Ellipt
 

(1). If you have a nozzle, the flow can be steady and subsonic throughout the nozzle. In that case, the exit condition will affect the rest of the flow field. And the flow will be elliptic. (2). The same nozzle can also produce subsonic-transonic-supersonic flow, or even subsonic-transonic-supersonic-subsonic flow, depending upon the initial and the boundary cinditions. And it is well known that in the supersonic region, the flow can only affect the region in the downstream direction. This portion of the flow is hyperbolic. (3). In both cases, you are using the same governing equations.

Re: Type of PDE: Hyperbolic or Parabolic or Ellipt
 

The independent variables need to be identified in order to classify PDEs. If you consider space & time, the N-S equations are parabolic. If you drop viscous terms, they become hyperbolic. However, in space, they are elliptic (for subsonic flows) or hyperbolic (supersonic).

As pointed out, hyperbolic equations have characteristics. There are characteristic lines in spac e (like in the method of characteristics used for supersonic nozzle designs) and characteristics in space-time using which unsteady characteristic boundary conditions are designed for flux-split schemes. Therefore, identifying what independent variables do you want to base your classification on is important.

Re: Type of PDE: Hyperbolic or Parabolic or Ellipt
 

Simply speaking the answer to your question is because for the unsteady equations the eigenvalues can either be all positive (parabolic) if the flow is subsonic or some can be negative (hyperbolic) if the flow is supersonic. So in N-S solution parts of the flow field can be parabolic while parts can be hyperbolic. In fact you can have directions which are parabolic while other directions are hyperbolic.

In a parabolic equation (simple example is the heat equation) the value of the dependent variable (call it 'u') at a given time is dependent on the value of 'u' at previous times over the entire domain (or connected parabolic portion if the domain is mixed like in a transonic flow). In a hyperbolic equation (simple example is the wave equation) the value of 'u' at a given time is dependent on the value of 'u' at previous times from only a portion of the domain which is bounded by 'characteristics'.

If the unsteady terms are dropped the eigenvalues can either be complex (elliptic) or real (hyperbolic). The meaning of hyperbolic is slightly different now because there is no time. Again portions of a steady-state N-S solution can be elliptic while other parts are hyperbolic and the classifications can vary by direction.

In an elliptic equation (eg Laplace's eqtn) the value of 'u' at a point in the domain is dependent of the value of 'u' at all parts of the domain. For a hyperbolic equation (I can't think of a non-time dependent one of the top of my head, sorry) the value of 'u' at one point in the domain is dependent only the values of 'u' a limited part of the domain enclosed by characteristics.

The steady state situation is easier think about. I suppose you know about Mach Cones? Well if an aircraft were flying directly over you subsonically you could hear it before it got to you. However if it were flying supersonically you'd not hear it until some time after it passed over you i.e. not until you got into the Mach cone. the surface of the Mach cone represents the boundary of the characteristics coming from the airplane and you can't hear it until you're within this boundary.

The significance of all this for the Navier-Stokes equations is that a steady mixed hyperbolic-elliptic problem can be solved by leaving the time depedent terms in (or introducing 'fake' ones) and solving the equations time dependently until steady state is reached at which point you have your steady solution. Of course you can increase the rate of convergence to steady state by taking liberties with time dependent terms since you're interested in an unsteady solution. The reason for this trickery is that it is not generally easy or possible to solve mixed hyperbolic-elliptic problems

characteristics line
 

in studying CFD, partial differential equation that described by characteristics line to define the hole in physical problems, this characteristics are hyperbolic has two distinct solution, parabolic has one distinct solution, elliptic has no distinct solution and domain disturbance. now i need clear understanding about this idea and asking help or reference?

This is a mathematical aspect of a second order PDE (first order PDE are somehow always hyperbolic, they are not subject to a classification). You can find details in a book of mathematics or in some CFD texts. The key is the same, you consider the eigenvalues problem and see they are real and distint, real and coincident or complex.
Then, considering the physical problem governed by a certain PDE, the mathematical characterization gives insight about the physics and the BC.s that ensure the problem to be well posed.