[Shiv1510]

Hi everyone !

I am writing a finite difference explicit code to simulate a turbulent flow over flat plate using RANS equations. My closure model is k-epsilon, modified for wall as per Chien's model. I am having trouble understanding on how to calculate the wall shear stress (tau_w) which is used to calculate wall friction velocity which in turn further calculate u+ and y+. My thinking is to use simple 3rd order polynomial fit on the U velocities (Calculated from momentum equation) for 3 points above the wall to calculate du/dy. And then calculate wall friction velocity and u+ and y+. My knowledge of CFD and Turbulence is very limited. So it would be a great help if some one could guide me here please.

Any help is appreciated. Thank you !

[FMDenaro]

Originally Posted by Shiv1510

Hi everyone !

I am writing a finite difference explicit code to simulate a turbulent flow over flat plate using RANS equations. My closure model is k-epsilon, modified for wall as per Chien's model. I am having trouble understanding on how to calculate the wall shear stress (tau_w) which is used to calculate wall friction velocity which in turn further calculate u+ and y+. My thinking is to use simple 3rd order polynomial fit on the U velocities (Calculated from momentum equation) for 3 points above the wall to calculate du/dy. And then calculate wall friction velocity and u+ and y+. My knowledge of CFD and Turbulence is very limited. So it would be a great help if some one could guide me here please.

Any help is appreciated. Thank you !

The first help I can do is to ask you what you want simulate. Are you simulating a spatial evolving BL from laminar to transitional conditions or is the flat plate periodic?

For a fully turbulent flow, you have also analytical law for the wall turbulence

[Shiv1510]

Originally Posted by FMDenaro

The first help I can do is to ask you what you want simulate. Are you simulating a spatial evolving BL from laminar to transitional conditions or is the flat plate periodic?

For a fully turbulent flow, you have also analytical law for the wall turbulence

Hi FMDenaro, Thank you so much for the help. So my code is unsteady for laminar case, and I wish to achieve same with turbulent conditions. But for now I am making it steady sate for turbulent conditions just to check if the model will work or not. My flat plate is not periodic. I am attaching a image of my inlet velocity vs time for your perusal. I wish to achieve this after my steady state model works. My Reynolds number for turbulence varies from 400000 to 600000.

Thank you again

[FMDenaro]

Originally Posted by Shiv1510

Hi FMDenaro, Thank you so much for the help. So my code is unsteady for laminar case, and I wish to achieve same with turbulent conditions. But for now I am making it steady sate for turbulent conditions just to check if the model will work or not. My flat plate is not periodic. I am attaching a image of my inlet velocity vs time for your perusal. I wish to achieve this after my steady state model works. My Reynolds number for turbulence varies from 400000 to 600000.

Thank you again

Still not clear what you are doing...


1) The velocity in time means you want to prescribe a uniform velocity in inlet of intensity that changes in time?
2) What means that your Re number varies from 4 to 6x10^5? The flow developing over a flat plate is not turbulent at such Re numbers.
3) There is no a unique Re number but it depends on the x position along the flat plate.
4) The developing BL if the aim is to simulate the transition from laminar to turbulent condition can never be simulated using RANS. You have to solve the 3D unsteady Navier-Stokes equations


In conclusion, what you could do is supposing that the inlet is already turbulent and fully developed at the condition of you last time in the plot. You can prescribe the law profile and solving the 2D steady RANS equations to check if the solution converges toward the theoretical one.

[Shiv1510]

Originally Posted by FMDenaro

Still not clear what you are doing...


1) The velocity in time means you want to prescribe a uniform velocity in inlet of intensity that changes in time?
2) What means that your Re number varies from 4 to 6x10^5? The flow developing over a flat plate is not turbulent at such Re numbers.
3) There is no a unique Re number but it depends on the x position along the flat plate.
4) The developing BL if the aim is to simulate the transition from laminar to turbulent condition can never be simulated using RANS. You have to solve the 3D unsteady Navier-Stokes equations


In conclusion, what you could do is supposing that the inlet is already turbulent and fully developed at the condition of you last time in the plot. You can prescribe the law profile and solving the 2D steady RANS equations to check if the solution converges toward the theoretical one.

Hello again sir,

My apologies. Yes to answer the first question,
1) I want to simulate unsteady case where my velocity at the inlet rises from 0 or a very small value to the velocity determined on the basis of the Reynold's number prescribed. This I want to simulate as per the function in my previous image fro unsteady case.
2) I am simulating in a square domain about 1inchx1inch (0.0254mx0.0254m) and my bottom boundary is the plate. My characteristic length for the Reynolds number is the length of plate and I back calculate my free stream velocity from it, which I prescribe at the inlet.
3) The profiles that I am trying to study are all the the outlet of the domain/plate.
4) I have two versions of my code, steady and unsteady. 2D Unsteady laminar cases works well, but currently I am trying to incorporate the k-epsilon model for steady state case. Simulating unsteady case is next step if the model works.
5) For the case I am trying to simulate, Steady state, my Reynolds number is 6x10^5. Yes I agree with you, currently at the inlet, it is already turbulent.

Here I am having trouble how to calculate y+ and u+ in Chien's model ? I am trying to avoid use of wall function and instead calculate wall shear stress using polynomial fit on the 3 points above my plate and then calculating y+ and u= from that predicted shear. I need some advice on this approach.

Thank you again. and my apologies again.

[FMDenaro]

Originally Posted by Shiv1510

Hello again sir,

My apologies. Yes to answer the first question,
1) I want to simulate unsteady case where my velocity at the inlet rises from 0 or a very small value to the velocity determined on the basis of the Reynold's number prescribed. This I want to simulate as per the function in my previous image fro unsteady case.
2) I am simulating in a square domain about 1inchx1inch (0.0254mx0.0254m) and my bottom boundary is the plate. My characteristic length for the Reynolds number is the length of plate and I back calculate my free stream velocity from it, which I prescribe at the inlet.
3) The profiles that I am trying to study are all the the outlet of the domain/plate.
4) I have two versions of my code, steady and unsteady. 2D Unsteady laminar cases works well, but currently I am trying to incorporate the k-epsilon model for steady state case. Simulating unsteady case is next step if the model works.
5) For the case I am trying to simulate, Steady state, my Reynolds number is 6x10^5. Yes I agree with you, currently at the inlet, it is already turbulent.

Here I am having trouble how to calculate y+ and u+ in Chien's model ? I am trying to avoid use of wall function and instead calculate wall shear stress using polynomial fit on the 3 points above my plate and then calculating y+ and u= from that predicted shear. I need some advice on this approach.

Thank you again. and my apologies again.

There are many contractidion in the approach you are using for this flow problem. You are assuming that the inlet averaged profile is a function of time, that is Uinf=U(0,y,t). That means you need to use the URANS formulation, not the RANS. And using URANS you should work with a 3D formulation as is now largely addressed in literature.
Since you are starting from a fully developed flow, the Re number must be based on a lenght that is characteristic of your flow problem and that lenght is not the lenght of the flat plate. You should use the BL characteristic lenght. Note that your time-dependent inlet will change in time the Re number. Finally, y+ is a function, is nothing else but the local reynolds number measured using the distance from the wall. You need to compute dynamically the wall stress to compute u_tau and then the function y+.

[Shiv1510]

Originally Posted by FMDenaro

There are many contractidion in the approach you are using for this flow problem. You are assuming that the inlet averaged profile is a function of time, that is Uinf=U(0,y,t). That means you need to use the URANS formulation, not the RANS. And using URANS you should work with a 3D formulation as is now largely addressed in literature.
Since you are starting from a fully developed flow, the Re number must be based on a lenght that is characteristic of your flow problem and that lenght is not the lenght of the flat plate. You should use the BL characteristic lenght. Note that your time-dependent inlet will change in time the Re number. Finally, y+ is a function, is nothing else but the local reynolds number measured using the distance from the wall. You need to compute dynamically the wall stress to compute u_tau and then the function y+.

Thank you again sir,

As I mentioned before my knowledge of CFD and turbulence is limited. I did not that URANS is different. I would like to correct my self and explain that I am using dk/dt and de/dt for unsteady cases. I hope my understanding is right about URANS, that it has an additional time derivative term than RANS equation (k-epsilon equations).
I am sorry, I did not understand the BL characteristic length ? Is it the boundary layer thickness ? but that is what I am trying to predict.

[FMDenaro]

Originally Posted by Shiv1510

Thank you again sir,

As I mentioned before my knowledge of CFD and turbulence is limited. I did not that URANS is different. I would like to correct my self and explain that I am using dk/dt and de/dt for unsteady cases. I hope my understanding is right about URANS, that it has an additional time derivative term than RANS equation (k-epsilon equations).
I am sorry, I did not understand the BL characteristic length ? Is it the boundary layer thickness ? but that is what I am trying to predict.

Yes, URANS is the formulation with the time derivatives in the equations. That assumes a different meaning of the averaging and must be described by a 3D formulation, not in 2D. However, try the 2D case. Be aware that, after you have reached the time where the inflow velocity does no longer change, the solution has to converge to a steady state, that is the time derivatives will vanish.
Yes, the characteristic lenght is the boundary layer thickness. Actually you are not predicting it in your simulation since you inflow turbulent profile has the in-built boundary layer thickness...

[Shiv1510]

Originally Posted by FMDenaro

Yes, URANS is the formulation with the time derivatives in the equations. That assumes a different meaning of the averaging and must be described by a 3D formulation, not in 2D. However, try the 2D case. Be aware that, after you have reached the time where the inflow velocity does no longer change, the solution has to converge to a steady state, that is the time derivatives will vanish.
Yes, the characteristic lenght is the boundary layer thickness. Actually you are not predicting it in your simulation since you inflow turbulent profile has the in-built boundary layer thickness...

I see, ok, but in my case at the inlet boundary all the nodes in transverse direction (Except one on the plate/bottom boundary) are prescribed constant velocity (U_inf) so its not fully developed. As per your suggestions, I should prescribe a fully developed profile at the inlet and from that profile I find the the in-built boundary layer thickness and use it to find my Reynold number ? Do I have to change my boundary conditions on P at the exit as well ? Currently I am prescribing P_atm at the outlet. Also could you guide me as to where can I find such formula/or method to find function for a fully developed turbulent profile ?

Thank you again so much sir

[FMDenaro]

Originally Posted by Shiv1510

I see, ok, but in my case at the inlet boundary all the nodes in transverse direction (Except one on the plate/bottom boundary) are prescribed constant velocity (U_inf) so its not fully developed. As per your suggestions, I should prescribe a fully developed profile at the inlet and from that profile I find the the in-built boundary layer thickness and use it to find my Reynold number ? Do I have to change my boundary conditions on P at the exit as well ? Currently I am prescribing P_atm at the outlet. Also could you guide me as to where can I find such formula/or method to find function for a fully developed turbulent profile ?

Thank you again so much sir

Sorry to say that you have no knowledge of what you are doing...
If you prescribe at inlet a uniform velocity profile that is not a fully developed turbulent profile but is the inviscid (external) velocity field that is prescribed in the standard theory of laminar boundary layer. That is totally different problem and cannot be simulated by RANS/URANS but must be described but a DNS formulation. Then, assuming that the boundary layer evolves from the inviscid condition, when you reach the end of the flat plate you have not reached the Re_x value for a turbulent field but only transitional condition.
I strongly suggest to stop thinking this as a CFD problem and coming back to the fundamental of fluid dynamics of BL theory.