I've recently began studying transonic flow and have been reading up on numerical methods (finite difference) to solve such problems.

I have heard that a shock (supersonic-subsonic) is a viscous phenomena - it is possible in the navier stokes equations due to the presence of the viscosity term. It can also appear in the finite difference solutions of the Euler equations due to the use of artificial viscosity when constructing our finite difference stencil.

I'm not sure why viscosity is necessary for a shock to occur. If someone could explain this I'd be very grateful.

Cheers, Dave

It's not. It's an inviscid phenomenon. I recommend reading any gasdynamics textbook.

I have another question then.

Can this inviscid phenomenon, or even shocks, accrues in incompressible non viscous two-phase flow?

I have been reading gas dynamics textbooks - they are what prompted the question.

I am studying inviscid flow around an airfoil, the boundary conditions are 1. No flow through the airfoil (impermeability) 2. Far-field boundary conditions (steady in my case) 3. Wake condition (wake can't bear aerodynamical loads) 4. Kutta condition (trailing edge)

In addition to these you need an entropy condition - the navier stokes eqns rule out expansion shocks because of the the presence of the viscous term. The euler equations don't include viscosity so an additional, entropy, condition is required.

In finite difference equations I've read that it's only possible to solve flows for shocks if we incorporate artificial viscosity into the equations. This is what I don't understand.

Shocks obviously aren't exclusively an invicid phenomenon - they occur in real life and the thickness of the shock is related to the viscosity (J. Moran). D

The artificial viscosity is only required to stabilize the flow algorithm. There are algorithms that use explicitly added artificial damping, and there are algorithms that accomplish the same thing in more subtle ways, such as limiters in high-resolution schemes. In transonic flow around an airfoil, a shock arises because the pressure waves generated at the trailing edge travel upstream at the speed of sound, but the since the flow traveling over the airfoil downstream is at or above the local speed of sound, the waves "pile up" at a point, resulting in a shock wave - actually a very small region across which a large gradient occurs.

Thanks ag - that's a very useful explanation. I hadn't really thought of shock generation in terms of sound waves - I can see though that the pile-up would give a region of discontinuity.

Cheers, D

Hi,

I am very sure that shock generation is an VISCOUS phenomenan. I am sure that this is explained in some book, currently i dont rember that book name but I will get back soon with the refernce.

Dave is absolutely wright that shock is caputred by Euler equations only because of numerical viscosity....

Thanks, Vinayender.

Actually, shocks form as the flow tries to satisfy boundary conditions that result in necessary changes occurring over very small regions. In some situations the presence of viscosity may have an effect, but shock formation does not require viscosity. The shock that sets up in the diverging region of a DeLaval nozzle has nothing to do with the presence of viscosity and everything to do with the flow needing to adjust to the exit pressure.

And following on ag's explanation, once the flow is supersonic the only way to decelerate it is through a shock.

Why can't the flow decelerate smoothly?

I know that shocks decelerate the flow - I just don't understand why some smooth machanism of deceleration isn't possible.

Cheers, D

Dave,

The answer has alredy been given to you. The reason is in "piling up" of pressure waves that travel with the speed of sound. Just imagine supersonic flow impinging on a disk. Obviously, it has to decelerate down to zero on the disc surface. In order for the deceleration to happen smoothly, the information has to propagate upstream, so that the flow properties (pressure) can adjust accordingly. However, no information in the flow can move faster than the speed of sound; therefore, a sudden change is the only way.

cc

Hi,

the origin of shocks has absolutely nothing to do with viscosity. It can mathematically be shown that they occur in one dimensional compressible inviscid flows (Riemann's theory). These shocks are real jumps, i.e., they are infinitely thin.

Viscosity and heat conduction lead to smooth profiles of the flow quantities across the shock although the thickness of the shock is so small that it remains a 'jump' from a macroscopic (and numerical) point of view.

The theory of shocks with viscosity and heat conduction dates back to the work of Richard Becker from 1922.

The fact that viscosity leads to differentiable shock profiles has suggested the concept of artificial viscosity (John v. Neumann).

Regards, Markus

Dear Dave,

It is easy to understand the origin of shock if you consider a simple non-linear equation, such as the inviscid Burger's equation. For any given profile, you can see that the characteristics have a slope equal to 1/u, and therefore are non-parallel straight lines. Accordingly, there are portions of the profile that flattens off and portions that steepen. (This is referred to as non-linear wave steepening, you can see this for yourself if you draw it on a piece of paper; also there in LeVeque's book, if I remember right). All physical processes are a balance in nature, and the wave steepening (which increases the gradients) is opposed by viscous effects (which smears off the gradients). The resulting equilibrium is shock. Physically, a shock cannot occur in the absence of viscosity. However, mathematically shocks are solutions to the Euler equations (these happen at the vanishing viscosity limit). And the only reason why you get shock as a solution in a numerical Euler computation is the presence of numerical viscosity from the numerical discretization that opposes wave steepening and stabilizes the procedure.

Hope this helps

Regards,

Ganesh

I Completely agree with you GANESH Bhaii....

Thanks, Vinayender.

Sorry Ganesh, I do not understand your explanation. Perhaps some references to books or articles would help.

My understanding of shock waves is similar to what Dave wrote - in a 1D compressible fluid there are three characteristic waves traveling at U, U+c and U-c (where U is fluid velocity and c is speed of sound).

When U > c, all the characteristic waves are traveling in the same direction, the pressure waves emanating from the original disturbance "pile up" into a front or better known as the shock.

I do not see how viscosity enters and your Burger's equation analogy seems to contradict what you say - Burger's equation does not contain a dissipative term (second derivative) only a convective term.

I assume what he means is that viscosity forces the solution to be single valued (otherwise in the case of the inviscid Burgers equation there could 3 different choices for the velocity at a single point). Viscosity allows the solution to remain single valued with the correct jump conditions then arisising from the zero viscosity limit (e.g. Whitham's book on nonlinear waves). This is a good example of a small term (the viscosity) having a finite effect in the zero viscosity limit.

Tom's response is along the lines of the interpretation that I have always learned - basically, viscosity doesn't make the shock, it just makes it physically plausible. Following Whitham's book (Linear and Nonlinear Waves) breaking of the solution leads to a triple-valued function in compressive regions. This is of course nonsense in the real world, and in those regions where gradients get very large the viscosity and heat conduction become very important and lead to a continuous solution (as noted by ganesh). But the mechanism by which the shock is initiated is independent of viscosity, and we can deal with shocks on a mathematical basis by replacing the triple-valued solution with a discontinuity.

Hi,

My Aurguments are as follow,

Do you all agree tbat there is decrease in Total Pressure across the shock(which represents losses). If Shock formation is an Invisid phenomenan, how does this loss takes place ?

All this losses occur in the physical thin region (A physical Shock) and these losses are purly because of viscous effects or else how can a loss takes place ?

For all practicaly purposes, what happens in this thin regoin is not important and hence we mathematically model the shock as a discontinutity and dumping all tbe viscous effects in the discontinutity AND Hence the rest of the flow field can be treated as Invisid.

But there are some physical problems where we cant treat shock as a discontinutity, such as in some combustion problems the time scales involved are very small that when the combustion particles passes through the shock layer, significent reactions can occur. In such we need to study the internal structure of the thin VISCOUS shock layer. And in all other cases we can comfortabely dump our ignorance of the VISCOUS shock layer by a discontinututy and study the rest.

And CFD can capture a shock even in invisid computations only because of numerical dissipation.

"Is Viscocity is the only reason for losses in the fluid flow that occur?" I guess yes it is .. Any comments please..

Thanks, Vinayender.

Hi,

the total pressure loss across a shock wave can be calculated from the (inviscid) Rankine-Hugoniot equations. It is a consequence of the non isentropic character of the process.

Regards, Markus

However this implies a loss (dissipation) which is absent in the invisicd equations. As I implied above the Rankine-Hugonoit relations naturally arise from the limit of zero viscosity of the viscous solution (just like the circulation does in the Kutta condition).

The process leading up to the shock is inviscid while the jump conditions across the shock are viscous.