Turbulent Boundary Layer Equation?

aerosjc · January 11, 2017, 19:27

Hi,

I'm reading a paper involving turbulent boundary layer equation. And I can not understand the details of the deduction.

For the turbulent boundary layer, the governing equation can be simplified as follows,

$\frac{\partial}{\partial x_{2}} (\nu + \nu_{t}) \frac{\partial u_{i}}{\partial x_{2}} = F_{i}, i = 1, 3$

where $F_{i} = \frac{1}{\rho} \frac{\partial p}{\partial x_{i}} + \frac{\partial u_{i}}{\partial t} + \frac{\partial}{\partial x_{j}} u_{i} u_{j}$

$F_{i}$ can be regarded as constant along normal-wall direction. Then we can integrate the above equation as ODE along normal-wall direction, with the boundary condition at the wall and the edge of the boundary layer. That makes sense. However, the result in this paper is what I can not get, especially $\int^{\delta}_{0} \frac{y dy}{\nu+\nu_{t}}$ .

Screenshot of the paper is as follows.

FMDenaro · January 12, 2017, 03:54

but x2=0 is exactly the wall or is the value where the stress wall condition is imposed outside of the wall? In general, LES can or not resolve the boundary layer depending on the grid resolution

sbaffini · January 12, 2017, 04:27

May this deduction be of some help?

https://www.cfd-online.com/Forums/bl...on-part-1.html

The approach is identical, but the equations are solved exactly in + variables.

aerosjc · January 12, 2017, 12:29

Great! It helps a lot! Thank you!

And I read your post about the general wall function. I'm trying to include the pressure gradient effects in hybrid LES/RANS method. Your post helps a lot. And I guess the Reichardt function in your approach can be replaced with a more precise function.

Current approach I use in my codes is to adopt the turbulent viscosity given by S-A model and then substitute it into the TBLE equation with F_u = 0 to solve the ODE to get an analytical profile for u^+. It matches with DNS data.

Employ the above profile in your approach, I guess the simulation would be better for separated flow.

What's your opinion about the above method? And if I use the turbulent viscosity profile by S-A model and in the meantime include the pressure gradient, will there be conflicts between S-A model and the pressure gradient modeled by another approach?

aerosjc · January 12, 2017, 12:31

Sorry.

x2 = 0 is the wall.

I agree with you that LES can or not resolve wall boundary layer depending on the grid. That's why I turn to wall model.

aerosjc · January 12, 2017, 14:02

Oh! I read your following parts. And you improve the velocity profile. But regarding my approach, I concern

.

In the current approach I adopt,

has already been obtained by substituting turbulent viscosity profile into TBLE equation to solve the ODE. At each iteration, I get solutions at the first off-wall nodes, $(y, u, \mu, \rho)$ . I provide these variables to a nonlinear equation solver to get

. The equation I solve is $u^+ y^+ = u y / \nu$ . Then I can add turbulent viscosity into the physical viscosity to get the efficient viscosity for LES simulation. The wall model I use is in fact to supply the efficient viscosity to LES. And all flow field is simulated by LES.

The pressure gradient breaks the above logic, since it's not analytical. If the velocity profile to be analytical(true without the pressure gradient), the nonlinear equation solver works normally. Of course, the approach is in essence numerical and it works with additional steps to compute

. We use $(y, y^+, \rho, \mu)$ to get $u_{\tau}$ .

$y^+ = \frac{y \rho u_{\tau}}{\mu}$

Then we use $(\frac{\partial P}{\partial x}, \mu, \rho, u_{\tau})$ to get

.

$F^+_u = \frac{\partial P}{\partial x} \frac{\mu}{\rho^2 u^3_{\tau}}$

The nonlinear equation solver also works.

However, the assumption that

is const is suitable? From the above procedure,

seems not constant. It's influenced by the solutions at each iteration.

So, what's your opinion about it?

sbaffini · January 13, 2017, 05:21

I don't know if I got exactly what you meant, so let me clarify what I did and what I do with that procedure.

1) I used to solve the equation (1) in the paper excerpt of your first post in a code I was working on. The equation was treated as a regular ODE (solved with the Thomas tridiagonal algorithm). The turbulent viscosity was based on the Van Driest mixing length (so, independent from the turbulent viscosity of the outer turbulent model and from any possible forcing). The nonlinearity was handled iteratively: given a first guess for $u_\tau$ , this was used to compute the Van driest turbulent viscosity, than a solution obtained, a new $u_\tau$ computed, and so on, for a certain number of iterations or until a certain reduction in the relative change of $u_\tau$ was achieved.

2) The procedure was not implemented by me and, in my opinion, it had some issues. First of all, we were not actually using any forcing, so using it in place of more standard wall functions was an overkill. Moreover, as no forcing was used, there was no point in solving along two directions, as the only needed one is that along the external velocity direction.

3) In a relatively long effort to solve at least some of the problems mentioned above i finally got to the procedure i wrote about in my blog: a wall function formulation (i.e., no ODE) valid for all the forcings and also for the thermal problem with arbitrary Pr-Pr_t numbers. In theory, if forcings are considered for the momentum, the formulation should be used in the two perpendicular directions parallel to the wall, one along the external velocity and the other perpendicular to it. As mentioned in part 3 of my posts, the main limitation of the formulation is that it considers a turbulent viscosity which does not feel the underlying velocity, which might be wrong in the case forcing is present.

4) I first implemented the procedure without forcing using a Newton-Rhapson procedure to solve for $u_\tau$ , and I'm currently using it in a full code, both for velocity and temperature, without any concern.

5) However, as soon as you start introducing the forcing, things get far more complicated, either if you work on the original ODE or even more so if you work on the implicit wall function.

6) Solving the original ODE with the forcing is not, per se, complicated. However, it turned out that convergence could not be achieved for all the forcings; above some extreme values (at the moment I don't remember if it was for positive or negative with respect to the main flow) the iterations got into some limit cycle without ever achieving convergence.

7) I found solving the implicit wall function with forcings even more complicated as my unknown, $u_\tau$ , has no information on the flow direction, which can invert (at the wall, with respect to the external one) in case of adverse pressure gradients. In practice, i converted the formulation to something of the form $Re = f\left(y^+\right)$ , where Re is signed and accounts for the inversion. I solved this graphycally, and it turns out that, just as for the ODE, not all the values have a solutions while for some values there are more than a solution.

8) Devising the Newton-Rhapson procedure for the case with the gradient was not easy at all. As you have noticed, the forcing term is also dependent from $u_\tau$ , and from my tests this had to be included. My final procedure looks like the following:

DO WHILE (ABS(utau-utaunew).GT.tol1*ABS(utau).AND.iter.LT.maxiter)
utau = utaunew
IF (ABS(utau).LT.tol2) EXIT
iter = iter + 1
yplus = y*ABS(utau)/nu
fplus = nu*f/(rho*ABS(utau)**3)
CALL profile(yplus,fplus,uplus,dFdutau)
num = uext - utau*uplus
den = dFdutau
utaunew = utau - num/den
ENDDO

Where uplus is the wall function value for the given yplus and fplus,

dFdutau = -(uplus+yplus*duplusdyplus-3*fplus*(yplus*fp-gp))

and fp and gp are the functions f and g cited in my blog (duplusdyplus is trivial and is also cited, it is the original function before the integration).

Note that the function F is the one for which the zero is seeked for in the Newton-Rhapson cycle, and is given by:

F = uext - utau*uplus

Also note that the derivative is w.r.t. utau, but most terms contain ABS(utau).

9) In conclusion, at the moment i am using the procedure above without forcings for both the velocity (1 direction, as there is no forcing) and temperature. If the forcing is used, i can get convergence only for low absolute values (low w.r.t. what? i still have to understand); otherwise, I either don't get convergence at all or I get under the proper choice of the starting value for $u_\tau$ . Note that the procedure above can return negative $u_\tau$ values, meaning that an inversion is present.

10) I'm pretty confident in the procedure without forcing; far from condifence for the forcing. More generally speaking, I had no access to some relevant references in the field of pressure gradient sensitized wall functions, so my approach might be intrinsecally wrong.

11) This leads to the first important question: is the forcing needed and/or is it really constant with respect to

? I found some controversy on this matter. On one side, Popovac and Hanjalic:

http://scholar.google.com/citations?...J:u-x6o8ySG0sC

show that convective and gradient terms tend to be constant and non null along

. Still, i have also read (can't find the reference now) that using only one of these terms may not be the correct approach. Indeed, as you can see from fig. 2 in the paper above, if one is present, the other is likely to be too, and both are somehow in equilibrium, making the forcing less important than if only one of them is considered (which contradicts the current main stream use of only the pressure gradient in wall function formulations).

12) Another question is on the use of a certain form of turbulent viscosity in all the cases. There are indeed people using a different turbulent viscosity for each external turbulence model. As also stated by Popovac and Hanjalic:

We close this section with a note regarding the issue of consistency of the wall boundary conditions with the full model equations solved in the outer flow region, raised by Kalitzin et al.[

5]. It is noted that the here presented GWFs for velocity and temperature are consistent with the outer model (account for convection, pressure gradient and other source terms) when the first near-wall grid node lies in the fully turbulent region. And so are the boundary conditions for all variables when the first grid point lies very close to the wall within the viscous sublayer. In the intermediate (buffer) region, this consistency is not fully ensured because the expressions containing blending functions may not satisfy exactly the full model equations. But this deficiency is pertinent to most known wall treatments, (including the standard wall functions and various blending methods), and seems unavoidable if the procedure of defining wall boundary conditions is to be simple, computationally robust, and easily incorporable into existing CFD codes. Wall treatment in form of wall functions is an approximation in which a compromise is sought between the simplicity and robustness on one side and the overall quality of performance in a variety of complex flows using different computational grids.

January 11, 2017, 19:27	Turbulent Boundary Layer Equation?	#1
aerosjc Member Jingchang.Shi Join Date: Aug 2012 Location: Hang Zhou, China Posts: 78 Rep Power: 13	Hi, I'm reading a paper involving turbulent boundary layer equation. And I can not understand the details of the deduction. For the turbulent boundary layer, the governing equation can be simplified as follows, $\frac{\partial}{\partial x_{2}} (\nu + \nu_{t}) \frac{\partial u_{i}}{\partial x_{2}} = F_{i}, i = 1, 3$ where $F_{i} = \frac{1}{\rho} \frac{\partial p}{\partial x_{i}} + \frac{\partial u_{i}}{\partial t} + \frac{\partial}{\partial x_{j}} u_{i} u_{j}$ $F_{i}$ can be regarded as constant along normal-wall direction. Then we can integrate the above equation as ODE along normal-wall direction, with the boundary condition at the wall and the edge of the boundary layer. That makes sense. However, the result in this paper is what I can not get, especially $\int^{\delta}_{0} \frac{y dy}{\nu+\nu_{t}}$ . Screenshot of the paper is as follows. Last edited by aerosjc; January 11, 2017 at 19:31. Reason: Format

January 13, 2017, 05:21		#7
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,152 Blog Entries: 29 Rep Power: 39	I don't know if I got exactly what you meant, so let me clarify what I did and what I do with that procedure. 1) I used to solve the equation (1) in the paper excerpt of your first post in a code I was working on. The equation was treated as a regular ODE (solved with the Thomas tridiagonal algorithm). The turbulent viscosity was based on the Van Driest mixing length (so, independent from the turbulent viscosity of the outer turbulent model and from any possible forcing). The nonlinearity was handled iteratively: given a first guess for $u_\tau$ , this was used to compute the Van driest turbulent viscosity, than a solution obtained, a new $u_\tau$ computed, and so on, for a certain number of iterations or until a certain reduction in the relative change of $u_\tau$ was achieved. 2) The procedure was not implemented by me and, in my opinion, it had some issues. First of all, we were not actually using any forcing, so using it in place of more standard wall functions was an overkill. Moreover, as no forcing was used, there was no point in solving along two directions, as the only needed one is that along the external velocity direction. 3) In a relatively long effort to solve at least some of the problems mentioned above i finally got to the procedure i wrote about in my blog: a wall function formulation (i.e., no ODE) valid for all the forcings and also for the thermal problem with arbitrary Pr-Pr_t numbers. In theory, if forcings are considered for the momentum, the formulation should be used in the two perpendicular directions parallel to the wall, one along the external velocity and the other perpendicular to it. As mentioned in part 3 of my posts, the main limitation of the formulation is that it considers a turbulent viscosity which does not feel the underlying velocity, which might be wrong in the case forcing is present. 4) I first implemented the procedure without forcing using a Newton-Rhapson procedure to solve for $u_\tau$ , and I'm currently using it in a full code, both for velocity and temperature, without any concern. 5) However, as soon as you start introducing the forcing, things get far more complicated, either if you work on the original ODE or even more so if you work on the implicit wall function. 6) Solving the original ODE with the forcing is not, per se, complicated. However, it turned out that convergence could not be achieved for all the forcings; above some extreme values (at the moment I don't remember if it was for positive or negative with respect to the main flow) the iterations got into some limit cycle without ever achieving convergence. 7) I found solving the implicit wall function with forcings even more complicated as my unknown, $u_\tau$ , has no information on the flow direction, which can invert (at the wall, with respect to the external one) in case of adverse pressure gradients. In practice, i converted the formulation to something of the form $Re = f\left(y^+\right)$ , where Re is signed and accounts for the inversion. I solved this graphycally, and it turns out that, just as for the ODE, not all the values have a solutions while for some values there are more than a solution. 8) Devising the Newton-Rhapson procedure for the case with the gradient was not easy at all. As you have noticed, the forcing term is also dependent from $u_\tau$ , and from my tests this had to be included. My final procedure looks like the following: DO WHILE (ABS(utau-utaunew).GT.tol1ABS(utau).AND.iter.LT.maxiter) utau = utaunew IF (ABS(utau).LT.tol2) EXIT iter = iter + 1 yplus = yABS(utau)/nu fplus = nuf/(rhoABS(utau)*3) CALL profile(yplus,fplus,uplus,dFdutau) num = uext - utauuplus den = dFdutau utaunew = utau - num/den ENDDO Where uplus is the wall function value for the given yplus and fplus, dFdutau = -(uplus+yplusduplusdyplus-3fplus(yplusfp-gp)) and fp and gp are the functions f and g cited in my blog (duplusdyplus is trivial and is also cited, it is the original function before the integration). Note that the function F is the one for which the zero is seeked for in the Newton-Rhapson cycle, and is given by: F = uext - utauuplus Also note that the derivative is w.r.t. utau, but most terms contain ABS(utau). 9) In conclusion, at the moment i am using the procedure above without forcings for both the velocity (1 direction, as there is no forcing) and temperature. If the forcing is used, i can get convergence only for low absolute values (low w.r.t. what? i still have to understand); otherwise, I either don't get convergence at all or I get under the proper choice of the starting value for $u_\tau$ . Note that the procedure above can return negative $u_\tau$ values, meaning that an inversion is present. 10) I'm pretty confident in the procedure without forcing; far from condifence for the forcing. More generally speaking, I had no access to some relevant references in the field of pressure gradient sensitized wall functions, so my approach might be intrinsecally wrong. 11) This leads to the first important question: is the forcing needed and/or is it really constant with respect to ? I found some controversy on this matter. On one side, Popovac and Hanjalic: http://scholar.google.com/citations?...J:u-x6o8ySG0sC show that convective and gradient terms tend to be constant and non null along . Still, i have also read (can't find the reference now) that using only one of these terms may not be the correct approach. Indeed, as you can see from fig. 2 in the paper above, if one is present, the other is likely to be too, and both are somehow in equilibrium, making the forcing less important than if only one of them is considered (which contradicts the current main stream use of only the pressure gradient in wall function formulations). 12) Another question is on the use of a certain form of turbulent viscosity in all the cases. There are indeed people using a different turbulent viscosity for each external turbulence model. As also stated by Popovac and Hanjalic: We close this section with a note regarding the issue of consistency of the wall boundary conditions with the full model equations solved in the outer flow region, raised by Kalitzin et al.[* 5]. It is noted that the here presented GWFs for velocity and temperature are consistent with the outer model (account for convection, pressure gradient and other source terms) when the first near-wall grid node lies in the fully turbulent region. And so are the boundary conditions for all variables when the first grid point lies very close to the wall within the viscous sublayer. In the intermediate (buffer) region, this consistency is not fully ensured because the expressions containing blending functions may not satisfy exactly the full model equations. But this deficiency is pertinent to most known wall treatments, (including the standard wall functions and various blending methods), and seems unavoidable if the procedure of defining wall boundary conditions is to be simple, computationally robust, and easily incorporable into existing CFD codes. Wall treatment in form of wall functions is an approximation in which a compromise is sought between the simplicity and robustness on one side and the overall quality of performance in a variety of complex flows using different computational grids. aerosjc likes this.

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January 12, 2017, 03:54		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	but x2=0 is exactly the wall or is the value where the stress wall condition is imposed outside of the wall? In general, LES can or not resolve the boundary layer depending on the grid resolution

January 12, 2017, 04:27		#3
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,152 Blog Entries: 29 Rep Power: 39	May this deduction be of some help? https://www.cfd-online.com/Forums/bl...on-part-1.html The approach is identical, but the equations are solved exactly in + variables.

January 12, 2017, 12:29		#4
aerosjc Member Jingchang.Shi Join Date: Aug 2012 Location: Hang Zhou, China Posts: 78 Rep Power: 13	Great! It helps a lot! Thank you! And I read your post about the general wall function. I'm trying to include the pressure gradient effects in hybrid LES/RANS method. Your post helps a lot. And I guess the Reichardt function in your approach can be replaced with a more precise function. Current approach I use in my codes is to adopt the turbulent viscosity given by S-A model and then substitute it into the TBLE equation with F_u = 0 to solve the ODE to get an analytical profile for u^+. It matches with DNS data. Employ the above profile in your approach, I guess the simulation would be better for separated flow. What's your opinion about the above method? And if I use the turbulent viscosity profile by S-A model and in the meantime include the pressure gradient, will there be conflicts between S-A model and the pressure gradient modeled by another approach?

January 12, 2017, 12:31		#5
aerosjc Member Jingchang.Shi Join Date: Aug 2012 Location: Hang Zhou, China Posts: 78 Rep Power: 13	Sorry. x2 = 0 is the wall. I agree with you that LES can or not resolve wall boundary layer depending on the grid. That's why I turn to wall model.

January 12, 2017, 14:02		#6
aerosjc Member Jingchang.Shi Join Date: Aug 2012 Location: Hang Zhou, China Posts: 78 Rep Power: 13	Oh! I read your following parts. And you improve the velocity profile. But regarding my approach, I concern . In the current approach I adopt, has already been obtained by substituting turbulent viscosity profile into TBLE equation to solve the ODE. At each iteration, I get solutions at the first off-wall nodes, $(y, u, \mu, \rho)$ . I provide these variables to a nonlinear equation solver to get . The equation I solve is $u^+ y^+ = u y / \nu$ . Then I can add turbulent viscosity into the physical viscosity to get the efficient viscosity for LES simulation. The wall model I use is in fact to supply the efficient viscosity to LES. And all flow field is simulated by LES. The pressure gradient breaks the above logic, since it's not analytical. If the velocity profile to be analytical(true without the pressure gradient), the nonlinear equation solver works normally. Of course, the approach is in essence numerical and it works with additional steps to compute . We use $(y, y^+, \rho, \mu)$ to get $u_{\tau}$ . $y^+ = \frac{y \rho u_{\tau}}{\mu}$ Then we use $(\frac{\partial P}{\partial x}, \mu, \rho, u_{\tau})$ to get . $F^+_u = \frac{\partial P}{\partial x} \frac{\mu}{\rho^2 u^3_{\tau}}$ The nonlinear equation solver also works. However, the assumption that is const is suitable? From the above procedure, seems not constant. It's influenced by the solutions at each iteration. So, what's your opinion about it?