Tuesday, December 13, 2011

MrsDrPoe: Boundary Layer Equations

It's once again Thesis Tuesday on the blog, and today we'll look some more at external flow.

One of the most important considerations of external flow is this region around the body in the flow that is affected by viscous stresses.  This region, called a boundary layer, possesses a large velocity gradient as the velocity transitions from 0 at the wall (from the no-slip condition) to 0.99*Uinfinity at the top of the boundary layer.  Since tau = mu*d/dy(u), we know that it is because of this velocity gradient that viscous stresses are important!  Typically, the height (distance from the surface to the top of the boundary layer) is denoted delta.

Fluid flow in the boundary layer is governed by a special reduced form of the governing equations.  We can find these equations using simple scaling analysis.  We will begin with the continuity equation for flow over a flat plate:

d/dt(rho) + d/dx(rho*u) + d/dy(rho*v) + d/dz(rho*w) = 0

Reducing for incompressible, 2D flow:

d/dx(u) + d/dy(v) = 0

We can say that du scales with Uinfinity (Uinfinity is a representative x-velocity), dx scales with L, dv scales with vs, and dy scales with delta.   For continuity to be satisfied, these two scaled terms must be of the same magnitude.  Thus, setting them equal, we find that vs = U*delta/L.

Next we will look at the Navier-Stokes y-momentum equation (so we've made assumptions of Newtonian Fluid, steady, constant properties, laminar, incompressible) for 2D flow:

rho*(u*d/dx(v) + v*d/dy(v)) = -d/dy(p) + v*(d/dx(d/dx(v)) + d/dy(d/dy(v)))

Applying our scaling arguments as before with the exception of p scaling with rho*U*U: rho*U*vs/L, rho*vs*vs/delta, rho*U*U/delta, v*vs/(L*L), v*vs/(delta*delta).  If we substitute in the value for vs that we found from the continuity equation:

rho*U*(1/L)*(U*delta/L), rho*(U*delta/L)*(1/delta)*(U*delta/L), rho*U*U/delta, v*(1/(L*L))*(U*delta/L), v*(1/delta)*(U*delta/L)

We can make these terms dimensionless by multiplying both sides by delta/(U*U*rho): (delta*deta)/(L*L), (delta*delta)/(L*L), 1, (delta*delta)/(L*L), 1/Re, 1/Re.  If we then take the limit as delta/L (the boundary layer is VERY thin) -> 0 and Re -> infinity, we see that the only term that does not go to zero is the pressure term, d/dy(p).  From these arguments, we can see that d/dy(p) = 0, or the pressure does not vary significantly in the direction normal to the wall.

Finally, we will apply scaling arguments to the Navier-Stokes x-momentum equation (again already reduced for 2D flow): rho*(u*d/dx(u) + v*d/dy(u)) = -d/dx(p) + v*(d/dx(d/dx(u)) + d/dy(d/dy(u))).  The terms become: rho*U*U/L, rho*vs*U/delta, rho*U*U/L, v*U/L, v*U/delta.  Substituting our expression for vs: rho*U*U/L, rho*(U*delta/L)*U/delta, rho*U*U/L, v*U/L, v*U/delta and non-dimensionalizing by multiplying by L/(U*U*rho): 1, 1, 1, 1/Re, (L*L/delta*delta), 1/Re.  If we again take the limit as delta/L -> 0 and Re -> infinity, we can see that the fourth term goes to zero; however, it is unclear what the last term becomes.  We can examine three possibilities: a) it's less that 1, b) it's greater than 1, or c) it's equal to 1.  

For case (a), scaling allows us to eliminate this term (since it is insignificant compared to the terms that reduced to 1) leaving: rho*(u*d/dx(v) + v*d/dy(v)) = -d/dx(p).  This equation is true for potential flow (irrotational, no viscous stresses), but since we definitely have these stresses present in the boundary layer, this option cannot be true.  

For case (b), the fifth term is the most significant term in the momentum equation so it would become: 0 = v*d/dy(d/dy(u)).  While this equation accounts for viscous stresses, it is purely diffusive and incorrect (we can see that d/dy(u) changes along the plate in the x-direction and is not constant).  

So case (c) must be correct, which means our boundary layer x-momentum equation can be written: rho*(u*d/dx(u) + v*d/dy(u)) = -d/dx(p) + v*d/dy(d/dy(u)).

A few miscellaneous notes:

Our full set of boundary layer equations consists of: 

d/dx(u) + d/dx(v) = 0
rho*(u*d/dx(u) + v*d/dy(u)) = -d/dx(p) + v*d/dy(d/dy(u))
0 = d/dy(p)

The boundary conditions for these equations are: u(x,0) = 0 (no-slip), v(x,0) = 0 (no-slip), u(x,delta) = Uinfinity, and u(x0,y) = uin(y) (starting velocity profile).

Incidentally, for our scaling: delta/L scales with 1/sqrt(Re).  Also, just outside the boundary layer, Bernoulli's equation applies.


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