Implicit momentum source based on the wall shear stress

magnushaese · May 7, 2018, 10:12

Hello,
i want to achieve a momentum source term that is adjusted, based on the wall shear stress $\tau_{\tau}$ . The reference case is a 1-D flow of a turbulent channel with viscous walls and periodic inlet/outlet. The flow is driven by the pressure gradient and symmetric. The governing equation is
$0 = -\frac{1}{\rho}\frac{\partial P}{\partial x} + \frac{\partial}{\partial y}\left( (\nu + \nu_t) \frac{\partial U}{\partial y}\right),$
where $-\frac{1}{\rho}\frac{\partial P}{\partial x}$
is to be assed in order to obain a fixed $Re_{\tau} = u_{\tau} h/\nu.$
I want to get the solution in an iterative manner by starting with the approximation of (Grundestam, Olof, Stefan Wallin, and Arne V. Johansson. "Direct numerical simulations of rotating turbulent channel flow." Journal of Fluid Mechanics 598 (2008): 177-199.), which relates the pressure gradient to the wallshearvelocity $u_{\tau}=\sqrt{\tau_{\tau}}$ with
$-\frac{1}{\rho}\frac{\partial P}{\partial x} = \frac{u_{\tau}^2}{h}.$
NOTE again: this is only an approximation.
Searching for available code reveals the fvOptions framework with the pressureGradientExplicitSource function. Moreover, the postprocessing tool of wallShearStress helps for the calculation of wall shear stresses. I have merged them into a working new momentumSource.
The idea is to adjust the delta in pressure gradient, need to achieve a net pressure gradient to give prescribed $\tau_{\tau}$ , which follows the idea of pressureGradientExplicitSource.C. Here, i apply
$\Delta \left(-\frac{1}{\rho}\frac{\partial P}{\partial x} \right) = \text{relax} * \left( \frac{u_{\tau eff}^2}{h}-\frac{u_{\tau 0}^2}{h} \right).$
Here, $u_{\tau eff}$ is the numerical wall shear velociy, $u_{\tau 0}$ a presribed value, relax a relaxation factor.

SOLVING:
Anyhow, by using $u_{\tau 0}^2/h$ as initial value for the pressure gradient, the solution does not converge. The "hope" of $\Delta \left(-\frac{1}{\rho}\frac{\partial P}{\partial x} \right)$ converging to 0 is not met, instead, the pressure gradient correction rises continuously, even when relaxating it with very low values.
The calculation of wallShearStress does work correctly, i have verified this with the wallShearStress postprocessing utility.

I would appreciate comments on this approach and other ideas on adjusting the momentumSource.

mAlletto · June 27, 2018, 05:51

what kind of solver are you using. Maybe if you post the equation i can try to help you.

mAlletto · June 27, 2018, 06:03

maybe you could try:

The following code was used to generate the image you clicked on:

$\Delta \left(-\frac{1}{\rho}\frac{\partial P}{\partial x} \right) = \frac{u_{\tau 0}^2}{h} + \text{relax} * \left( \frac{u_{\tau 0}^2}{h} - \frac{u_{\tau eff}^2}{h} \right).$

in this way if the actual wall shear stress is to height the pressure gradient is reduced and if it is too low it is increased. Too heigh pressure gradient means in may understanding too heigh mass flow.

May 7, 2018, 10:12	Implicit momentum source based on the wall shear stress	#1
magnushaese New Member eco Join Date: Jan 2017 Posts: 5 Rep Power: 9	Hello, i want to achieve a momentum source term that is adjusted, based on the wall shear stress $\tau_{\tau}$ . The reference case is a 1-D flow of a turbulent channel with viscous walls and periodic inlet/outlet. The flow is driven by the pressure gradient and symmetric. The governing equation is $0 = -\frac{1}{\rho}\frac{\partial P}{\partial x} + \frac{\partial}{\partial y}\left( (\nu + \nu_t) \frac{\partial U}{\partial y}\right),$ where $-\frac{1}{\rho}\frac{\partial P}{\partial x}$ is to be assed in order to obain a fixed $Re_{\tau} = u_{\tau} h/\nu.$ I want to get the solution in an iterative manner by starting with the approximation of (Grundestam, Olof, Stefan Wallin, and Arne V. Johansson. "Direct numerical simulations of rotating turbulent channel flow." Journal of Fluid Mechanics 598 (2008): 177-199.), which relates the pressure gradient to the wallshearvelocity $u_{\tau}=\sqrt{\tau_{\tau}}$ with $-\frac{1}{\rho}\frac{\partial P}{\partial x} = \frac{u_{\tau}^2}{h}.$ NOTE again: this is only an approximation. Searching for available code reveals the fvOptions framework with the pressureGradientExplicitSource function. Moreover, the postprocessing tool of wallShearStress helps for the calculation of wall shear stresses. I have merged them into a working new momentumSource. The idea is to adjust the delta in pressure gradient, need to achieve a net pressure gradient to give prescribed $\tau_{\tau}$ , which follows the idea of pressureGradientExplicitSource.C. Here, i apply $\Delta \left(-\frac{1}{\rho}\frac{\partial P}{\partial x} \right) = \text{relax} * \left( \frac{u_{\tau eff}^2}{h}-\frac{u_{\tau 0}^2}{h} \right).$ Here, $u_{\tau eff}$ is the numerical wall shear velociy, $u_{\tau 0}$ a presribed value, relax a relaxation factor. SOLVING: Anyhow, by using $u_{\tau 0}^2/h$ as initial value for the pressure gradient, the solution does not converge. The "hope" of $\Delta \left(-\frac{1}{\rho}\frac{\partial P}{\partial x} \right)$ converging to 0 is not met, instead, the pressure gradient correction rises continuously, even when relaxating it with very low values. The calculation of wallShearStress does work correctly, i have verified this with the wallShearStress postprocessing utility. I would appreciate comments on this approach and other ideas on adjusting the momentumSource.

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June 27, 2018, 05:51		#2
mAlletto Senior Member Michael Alletto Join Date: Jun 2018 Location: Bremen Posts: 616 Rep Power: 16	what kind of solver are you using. Maybe if you post the equation i can try to help you.

June 27, 2018, 06:03		#3
mAlletto Senior Member Michael Alletto Join Date: Jun 2018 Location: Bremen Posts: 616 Rep Power: 16	maybe you could try: The following code was used to generate the image you clicked on: $\Delta \left(-\frac{1}{\rho}\frac{\partial P}{\partial x} \right) = \frac{u_{\tau 0}^2}{h} + \text{relax} * \left( \frac{u_{\tau 0}^2}{h} - \frac{u_{\tau eff}^2}{h} \right).$ in this way if the actual wall shear stress is to height the pressure gradient is reduced and if it is too low it is increased. Too heigh pressure gradient means in may understanding too heigh mass flow.