Boundary Layer of Laminar Flow over a Flat Plate
 

Hello CFD Online Community

I try to validate a compressible-flow Navier-Stokes solver. My test case is a steady, laminar flow over a flat plate (zero pressure gradient), where the thermodynamic and transport properties depend on temperature.

For an incompressible flow (low Mach number AND constant density), the BLASIUS and POHLHAUSEN self similarity solutions are commonly used to validate the velocity and temperature profiles, respecively.

My 1st question is: Does the BLASIUS and POHLHAUSEN solution hold for both boundary conditions; isothermal and adiabatic wall???

If the answer is YES, let's consider the isothermal wall. Assume a (uniform) wall temperature that (considerably) differs from the free stream temperature. Then the density should vary as well, since: rho = p/(R*T) ... with constant p and R. But the BLASIUS flow solution holds only for constant rho, right? (continuity and momentum equations assume constant rho) Therefore only with the adiabatic wall condition, the velocity profiles can be compared with BLASIUS (neglect viscous dissipation).

But the POHLHAUSEN thermal solution is based on an isothermal wall, and the dimensionless similarity temperature profile is defined as:

theta = (T - T_w)/(T_free - T_w)

Now, for Pr=1 the velocity profile is identical to the temperature profile (in non-dimensional form).

Is this consistent??? In my opinion only if the isothermal wall temperature does not differ too much from the free stream temperature. Otherwise, there is a dependency of the velocity profile on the temperature (in fact, the coupling is vice versa).

Please correct me if there are some errors in reasoning!!

Actually, I found in WHITE's "Viscous Fluid Flow" (Chapter 7) a more general equation for compressible, laminar boundary layer flow (which includes BLASIUS and FALKNER-SKAN solutions). It is based on the CROCCO-BUSEMANN relations.

So, if BLASIUS allows me to validate incompressible, laminar flows with adiabatic wall condition (or isothermal with small temperature difference only), then I will have to compare my velocity profile for the isothermal case (any temperature difference) with the "CROCCO-BUSEMANN"-solution.

Is this the common practice to validate compressible, laminar flows over a flat plate? Where can the CROCCO solutions be found?

I greatly appreciate any comment on that and I'm very interested in the way you would treat such a problem.

MANY THANKS IN ADVANCE!!

Re: Boundary Layer of Laminar Flow over a Flat Pla
 

I do not know what the common practice is. The Blasius solution allows you to validate incompressible laminar flow over a semi-infinite flat plate at zero incidence, with constant density throughout the flowfield. Thus, temperature does not enter into the picture.

For compressible laminar boundary layer flow for the same configuration, the special case of unity Prandtl number gives streamwise velocity and temperature profiles that are similar (with temperature being solely a function of streamwise velocity). However, the velocity profile does not match the Blasius. You should validate your compressible Navier-Stokes code against a solution to Crocco/Busemann for Prandtl number 1 or 0.7, adiabatic or isothermal wall. The exact solutions depend on the Mach number and (in the isothermal case) on the wall temperature, in addition to the usual dependence on the Reynolds number. According to Schlichting's Boundary Layer Theory book, you can find numerical solutions to the exact equations in papers by Hantzsche and Wendt, but these are in German. So unless you can read a little German or obtain a translation, you will have to look elsewhere for solutions. You might need to solve the Crocco/Busemann equations yourself. Since these are ODEs, it should be relatively straightforward to solve them using Mathematica or similar software.

Note that your Navier-Stokes solution will not agree with the Blasius or Crocco/Busemann solutions near the leading edge. The boundary layer theory does not account for the physics of the sharp leading edge, and the Navier-Stokes solver will also not resolve the leading edge without very special meshes. Only at about x=1 or further downstream can you expect good agreement.

Re: Boundary Layer of Laminar Flow over a Flat Pla
 

Thank you very much ANANDA!

I see, it can be understood what my problem is...

I will check SCHLICHTING's book and the publications of HANTZSCHE and WENDT. Fortunately, I speak German as well.

At this point I would also like to mention another paper on this issue: "Temperature and Velocity Profiles in the Compressible Laminar Boundary Layer with Arbitrary Distribution of Surface Temperature", 1949 by CHAPMAN and RUBESIN.

Let me repeat the conclusion: BLASIUS solution cannot be used for validation, when the wall is isothermal and its temperature is significantly different from the free stream temperature, since this has an impact on the density in the zero pressure gradient flow. If the density is not constant, the simplifications of the continuity and momentum equations (which lead to the BLASIUS ODE), is not possible. Hence, BLASIUS solution is not valid in this case, no matter if the Mach number is lower than approximately 0.3 (in my test case M = 0.05), but because my density is changing with temperature.

Does anyone agree on that?

By the way, in my first message I've assumed that POHLHAUSEN's solution is valid for isothermal wall. Is it also valid for an adiabatic wall condition?

In fact, this is a very interesting topic for me! I would like to know what the opinion and experience of other people is on this issue.

Comments are still highly appreciated!!

All the best!!

Re: Boundary Layer of Laminar Flow over a Flat Pla
 

ANANDA,

thank you also for the information about the boundary layer near the leading edge. Indeed, I observe some strange results near the singularity, which are most probably caused by numerics (large gradients). But this problem appears only local at the first few grid points near the leading edge. To mitigate this, I have clustered my grid in the flow direction as well, taking into account that the computational effort should not be stressed too much. Of course I will compare my velocity and temperature profiles further downstream.

The laminar boundary layer theory also presumes that the slenderness postulate is valid, which means d is much smaller than L or sqrt(Re_L) much larger than 1

cheers!

Re: Boundary Layer of Laminar Flow over a Flat Pla
 

Yes, I agree with your statement on the Blasius solution. Although the Blasius solution can be interpreted (by change of variable, see Howarth-Dorodnitzn or Illingworth-Dorodnitzn transformation) for some variable-density boundary layer, the transformation involves an unrealistic assumption about the viscosity. You would have to change your Navier-Stokes viscosity routine to match that, and also solve an auxiliary equation to invert the transformation of the Blasius solution. So I would say don't bother with it.

Integral techniques like Pohlhausen's could be used for variable density boundary layers, I suppose, including isothermal and adiabatic walls. To use the original method of solving first the momentum equation and then the energy equation, the temperature changes must not be too drastic. For more general compressible boundary layers, you could solve the y-integral momentum and energy equations in coupled fashion. The useful outcome is the x-variation of skin friction and heat transfer. However, I think you would be faced with the problem of assuming reasonable velocity and temperature profiles for the general case, and then having to validate your Pohlhausen technique for the general compressible case! Also, you do not obtain velocity/temperature profiles to compare with your Navier-Stokes solution, because the profiles are part of your input.

I would lean toward the Busemann/Crocco formulation, which will give you velocity/temperature profiles as well as skin friction and heat transfer. This seems to be general, direct and more reliable. For the particular case of interest to you (flat plate at zero incidence), this should reduce to coupled ODEs in the normal direction (for the general case with pressure gradient, it stays as coupled parabolic PDEs). As I said, once you extract the proper formulation from a paper or book, you should be able to solve the ODE system in a straightforward way using ODE subroutines or, even more convenient, software such as Mathematica. This would save you from having to scan and digitize plots from old papers, and would enable you to run exactly the conditions you want rather than the ones available in the papers. The paper by Chapman & Rubesin you mention was famous in its day.

Yes, the leading edge is hard to resolve, but luckily has little influence on the solution further downstream. Nontrivial exact solutions of the compressible Navier-Stokes equations are hard to come by. Good luck to you.

Blasius
 

Hi!. I must validate my flow with blasius. First I flared the ceiling so the flow doesn't accelerate. it still didn't match Blasius. I really don't know what to do. The freestream velocity is now decreasing. Let me know if you have any suggestions.

What sort of boundary condition are you using on your 'ceiling'?

bl
 

I am using free slip for the ceiling and the side walls of the domain

Getting the upper wall right so that you don't have a stream-wise pressure gradient sounds bit tricky. Have you tried an outflow or farfield bc on the ceiling? I've used a farfield bc in a similar verification case (comparing to Blasius) and it worked just fine.

Which is available for previewing as a reprint in Google books.

I will try this today. Thank you a lot for your reply.

Boundary layer
 

Hi!I tryied to specify the top surface as an opening with 0 Pa pressure. In Ansys Cfd there is no farfield option for boundaries types or I do not know how to specify one. Do you have other suggestions??

chapman paper
 

Does anyone have access to the Chapman paper?

"Temperature and Velocity Profiles in the Compressible Laminar Boundary Layer with Arbitrary Distribution of Surface Temperature", 1949 by CHAPMAN and RUBESIN

couldn't find it.

thanks :)