3D Tollmien-Schlichting Waves Imposed in a Channel Flow (Physics, etc)

Santiago · March 27, 2019, 05:38

Hi All,

So I am trying to do some further tests on a 2nd-order code Incompressible Navier Stokes equations, by studying transition to turbulence in a Poiseuille flow. Specifically, I'm interested to see whether the code is energy-conserving by modelling a 3d TS wave in a Poiseuille flow. Note that I have tested the code for a 2d inviscid Taylor vortex and the code holds. Now, I have calculated the perturbation velocities via the Orr-Sommerfeld/Squire equations, and I get the Eigenspectrum and corresponding Eigenfunctions (velocities/wall-normal vorticity) for the following Poiseuille flow:

$U^0 = 1-y^2,\,\, -1 \le y \le 1$

With a Reynolds number based on the centreline velocity and the channel's half-height $Re_{cl}=\frac{U_{max}(h/2)}{\nu}=5000$ . From here, the driving pressure gradient is just:
$\Pi = -\frac{2}{Re}$

For the linear perturbation I choose a streamwise and spanwise wavenumbers of $\alpha = 1.12, \beta= 2.1$ , respectively. For the eigenvectors, the most unstable mode has been chosen (that where Real(Lambda) is the closest to zero). All the results of this are attached as PDF's.

I have imposed the velocity/vorticity perturbations into my code, without success (meaning not getting the transition), so clearly there's something I'm not quite getting right. Now, my questions are:

Orr-Sommerfeld Modelling: Are my results obtained (meaning perturbation velocities/vorticity) reasonable? Are these correct?
Equations to solve for temporal transition modelling: In my opinion, the equations to be solved when studying spatio-temporal transitions are the incompressible Navier-Stokes equations for the perturbation velocities, that is:
$u_{i,t} +u_j u_{i,j} + u_j U^0_{i,j} + U^0_j u_{i,j} = -p_{,i}+\frac{1}{Re}u_{i,kk}+\Pi_i$

Which results from the splitting of the instantaneous velocity into base velocity and perturbation velocity, $\widetilde{u_i} = U^0_i + u_i$ in the INS, and then substracting the INS of the base flow. Now my questions are: Is the previous equation correct? I'm having doubts about the pressure gradient $\Pi_i$ . Second: Should be these the equations I should be solving for simulating temporal transitions? Or, Should I go just with the classic INSE for the poiseuille flow and superposed fluctuations?

FMDenaro · March 27, 2019, 11:45

Your goal is not very clear to me. What do you mean for checking the energy-conserving property by using the incompressible model of the NSE?
You do not solve any energy equation but you can only deduce the kinetic energy from the solution of the momentum equation. Furthermore, the kinetic energy equation is not in a conservative form if the fluid is viscous, the dissipation term acting.

Santiago · March 27, 2019, 12:25

Quote:

Originally Posted by FMDenaro

Your goal is not very clear to me. What do you mean for checking the energy-conserving property by using the incompressible model of the NSE?
You do not solve any energy equation but you can only deduce the kinetic energy from the solution of the momentum equation. Furthermore, the kinetic energy equation is not in a conservative form if the fluid is viscous, the dissipation term acting.

Hello Fillipo!

Well, as energy I meant kinetic energy. And probably 'conservation' is not the appropriate term, but 'preservation'. If I were to simulate an inviscid 2D Taylor vortex, the K. Energy will be 'preserved', or stay constant in an 'energy preserving' INSE solver...

... My objective with this test is to see whether the energy growth rate of a 2d TS perturbance is preserved, in a 2nd order sense. My problem is that I am not being able to run the case successfully, either because i have the input data (perturbation velocities) or the INSE solver for the perturbations wrong.

Note that i have tested my INSE solver against other cases: 2d taylor vortex, turbulent channel flow at Re_tau = 600, etc. My error here is more conceptual in this particular case.

FMDenaro · March 27, 2019, 12:37

The kinetic energy is indeed conserved only if an inviscid and non conductive fluid is assumed, that is seen by writing the integral of the kinetic energy equation over the whole volume and using Gauss. But your problem is that you need to evaluate the kinetic energy from the numericla solution of the momentum as you do not solve directly the KE equation.The only possible source of kinetic energy would appear if the divergence-free constraint is not satisfied as the term p*(div v) will enter in the production of kinetic energy even in inviscid flow.
But if your test on inviscid cases showed that the total kinetic energy remains constant in time I would esclude a problem due to accumulation of energy.

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March 27, 2019, 11:45		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	Your goal is not very clear to me. What do you mean for checking the energy-conserving property by using the incompressible model of the NSE? You do not solve any energy equation but you can only deduce the kinetic energy from the solution of the momentum equation. Furthermore, the kinetic energy equation is not in a conservative form if the fluid is viscous, the dissipation term acting.

March 27, 2019, 12:37		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	The kinetic energy is indeed conserved only if an inviscid and non conductive fluid is assumed, that is seen by writing the integral of the kinetic energy equation over the whole volume and using Gauss. But your problem is that you need to evaluate the kinetic energy from the numericla solution of the momentum as you do not solve directly the KE equation.The only possible source of kinetic energy would appear if the divergence-free constraint is not satisfied as the term p*(div v) will enter in the production of kinetic energy even in inviscid flow. But if your test on inviscid cases showed that the total kinetic energy remains constant in time I would esclude a problem due to accumulation of energy.