Validity & transformation of boundary layer equations

Obad · April 23, 2015, 15:26

Hi folks,

im currently trying to write a code to solve the 2D incompressible/compressible laminar boundary layer equations with a constant cross flow component over a flat plate with an infinite span.

Right now I'm struggling with transforming my equations to a coordinate system that is appropriate for this problem. I understand that using a rectangular grid is inefficient and hence a coordinate system is necessary that limits the computational domain so that more grid points are included in the boundary layer itself. I already read about a couple of transformation methods e.g. the Levy-Lees and Patankar-Spalding transformation.

So I have a couple of fundamental questions, that you might be able to answer.

So first of all, do these transformations fit the computational domain exclusively to the region of the boundary layer, so that the upper limit is exactly the boundary layer edge?

My second question is about the validity of the boundary layer equations outside the boundary layer. For example I could use a simple rectangular grid that is chosen big enough to include the entire boundary layer. So now a lot of the grid points would lie outside of the boundary layer. Since the boundary layer equations are tailored for solving the boundary layer, how would the solution outside of the boundary layer look like?
Would it be necessary to use PNS-equations to solve such a problem?

My third question is adressing the transformation methods. Some of them e.g. the streamfunction transformation and the Patankar-Spalding method make use of the streamfunction. I know how the streamfunction is defined in terms of the velocity components, but I don't get it how to define a function that describes the variation of the streamfunctions that cross the boundary layer edge.

Maybe someone out here in the forum can enlighten me

I appreciate your help!

robo · April 23, 2015, 16:20

Wouldn't the flow outside of the boundary layer be trivial? The outer flow is essentially inviscid, and it's just a flat plate, so it'd be constant velocity, constant pressure, would it not? I feel like I am misunderstanding the problem...

Obad · April 27, 2015, 03:59

Hi,

the flow is certainly not trivial outside the boundary layer. The flow outside the boundary is effected by the boundary layer itself, e.g. the streamlines will deflect upwards. Just imagine the flow would be supersonic, then there would be a shock wave at the leading edge and velocity, pressure etc. will change greatly.

The question is if the additional simplifications of the boundary layer equations compared to the PNS equations make a big difference?

For a 2D flow the difference is the y-momentum equation, which for the boundary layer diminishes to the pressure term only.

April 23, 2015, 15:26	Validity & transformation of boundary layer equations	#1
Obad Member Obad Join Date: Sep 2013 Posts: 42 Rep Power: 13	Hi folks, im currently trying to write a code to solve the 2D incompressible/compressible laminar boundary layer equations with a constant cross flow component over a flat plate with an infinite span. Right now I'm struggling with transforming my equations to a coordinate system that is appropriate for this problem. I understand that using a rectangular grid is inefficient and hence a coordinate system is necessary that limits the computational domain so that more grid points are included in the boundary layer itself. I already read about a couple of transformation methods e.g. the Levy-Lees and Patankar-Spalding transformation. So I have a couple of fundamental questions, that you might be able to answer. So first of all, do these transformations fit the computational domain exclusively to the region of the boundary layer, so that the upper limit is exactly the boundary layer edge? My second question is about the validity of the boundary layer equations outside the boundary layer. For example I could use a simple rectangular grid that is chosen big enough to include the entire boundary layer. So now a lot of the grid points would lie outside of the boundary layer. Since the boundary layer equations are tailored for solving the boundary layer, how would the solution outside of the boundary layer look like? Would it be necessary to use PNS-equations to solve such a problem? My third question is adressing the transformation methods. Some of them e.g. the streamfunction transformation and the Patankar-Spalding method make use of the streamfunction. I know how the streamfunction is defined in terms of the velocity components, but I don't get it how to define a function that describes the variation of the streamfunctions that cross the boundary layer edge. Maybe someone out here in the forum can enlighten me I appreciate your help!

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April 23, 2015, 16:20		#2
robo Member robo Join Date: May 2013 Posts: 47 Rep Power: 13	Wouldn't the flow outside of the boundary layer be trivial? The outer flow is essentially inviscid, and it's just a flat plate, so it'd be constant velocity, constant pressure, would it not? I feel like I am misunderstanding the problem...

April 27, 2015, 03:59		#3
Obad Member Obad Join Date: Sep 2013 Posts: 42 Rep Power: 13	Hi, the flow is certainly not trivial outside the boundary layer. The flow outside the boundary is effected by the boundary layer itself, e.g. the streamlines will deflect upwards. Just imagine the flow would be supersonic, then there would be a shock wave at the leading edge and velocity, pressure etc. will change greatly. The question is if the additional simplifications of the boundary layer equations compared to the PNS equations make a big difference? For a 2D flow the difference is the y-momentum equation, which for the boundary layer diminishes to the pressure term only.