CEE 262A H YDRODYNAMICS. Lecture 16 Boundary layers: Scaling and Blasius. Boundary layers over a solid surface (classical aerodynamics). Imagine a flow that has a velocity parallel to the surface far from the surface that varies as U E (s) where s is the distance along the surface.
Boundary layers: Scaling and Blasius
Imagine a flow that has a velocity parallel to the surface far from the surface that varies as UE(s) where s is the distance along the surface.
How does the boundary layer thickness, drag etc, depend on the flow and its variation?
We start with the 2D, steady Navier Stokes equations (written in our usual Cartesian coordinates) for a homogeneous fluid. If the curvature of the surface is small (and hence the flow is steady):
where x1 is the distance parallel to the surface and x3 is the distance normal to the surface. Note that we have used the perturbation pressure (hydrostatic pressure is removed).
Suppose that the freestream velocity ~ (written in our usual Cartesian coordinates) for a homogeneous fluid. If the curvature of the surface is small (and hence the flow is steady):U and the length scale for any flow variations is L, then a scaling for the momentum equations would be:
where the importance of viscous stresses is measured by the Reynolds number
I.e. if Re >> 1, viscous stresses can be neglected, at least in the outer (potential) flow.
Define inner- and outer-layer variables: (written in our usual Cartesian coordinates) for a homogeneous fluid. If the curvature of the surface is small (and hence the flow is steady):
u, w, x, z, p
Weak viscous stresses.
u+, w+, x+, z+, p+
Strong viscous stresses.
In order to make viscous stresses important in the inner layer, we want (assume all quantities are dimensionless)
Dimensionally, this implies that the vertical scale for the inner layer is d:
(z dimensional here)
we don't stretch x. So we have
Thus, continuity requires that
These could also have been obtained via nondimensionalizationwith
in powers of the small parameter
If we substitute these into the governing equations and look at the lowest-order terms:
Which give to lowest order the inviscid flow problem
Outer layer: variables:
We can complete the problem specification by requiring that the following asymptotic matching hold:
(outer limit of inner variable = inner limit of outer variable)
Since the inner vertical momentum equation implies the following asymptotic matching hold:
Then the matching condition requires
However, from the outer x-momentum equation, we know that at z=0
Thus, the pressure gradient inside the boundary layer is the same as
For the velocity the following asymptotic matching hold:
Thus, the pressure gradient in the boundary layer and the streamwise velocity that must be matched to are determined by the outer, potential flow, which is what determines UE(x).
Note that we require
The problem of boundary layers then becomes one in which the following asymptotic matching hold:
the pressure field is known and we only need to solve for
the developing velocity field:
Imposed pressure gradient.
This approach does not allow us to describe separated flow regions
Van Dyke “Album of Fluid Motion”
Uniform upstream flow
x = 0
In general, the pdes that describe bls are nonlinear and hence tough.
Like Stokes' first problem, there is no imposed vertical length scale. Can we find a similarity solution?
U the following asymptotic matching hold:E
Consider moving at speed UE and observing the growth of the
boundary layer in this moving frame that moves according to
x(t)=UEt. After some time t the frame has moved a distance
t=x/UE from the leading edge at x=0. In the moving frame, by
analogy to Stokes' first problem, the boundary layer grows roughly
as d0(t)=(nt)1/2. Substitution gives
Recall that the following asymptotic matching hold:
Then convert partials wrt x and z+ to derivatives wrth (let x+=x):
To convert the pdes to an ode using the similarity variable, we look at the momentum equation in terms of the streamfunction
Let’s choose the streamfunction as follows
Why this form?
The derivatives are given by we look at the momentum equation in terms of the streamfunction
So substituting into the momentum equation we get
Note that all the we look at the momentum equation in terms of the streamfunctionx factors cancel, so we end up with an ode after all:
Which is subject to the conditions that (noting that u0+=f ')
(match outer solution)
Unfortunately, this is not analytically integrable and so must be done numerically. This requires a “shooting” method – start at the wall and integrate out. Since we don’t have f’’, we must guess f’’ and see if the f’ tends to 1 far from the wall. The value at the wall is f''(0)=0.332.
The result looks like this: we look at the momentum equation in terms of the streamfunction
Schlichting “Boundary Layer Theory”
We found that the curvature of the velocity at the wall is determined by the pressure gradient
Thus, if the pressure gradient < 0, the boundary layer will be thin, whereas if pressure gradient > 0, the boundary layer will be thick. The first case is known as a favorable pressure gradient, whereas the second case is known as an adverse pressure gradient. This latter case leads to separation, e.g. for flow around a cylinder
Using the Blasius solution, we can define the boundary layer thickness more accurately, for example if we define the boundary layer by the point at which the velocity is 99% of the freestream velocity:
(note that this works for Rex< 105)
If we calculate the (dimensional) vorticity thickness more accurately, for example if we define the boundary layer by the point at which the velocity is 99% of the freestream velocity:
As expected, most of the vorticity is contained within the BL. In terms of vorticity (dimensional) we have
If we hadn’t accounted for u3 or the variation with height of u1, we would have repeated Stokes first problem with t replaced by x1/U .
What about the flux of vorticity from the wall?
Vorticity in the flow is introduced ONLY at the leading edge at x=0.
Thus, the skin friction coefficient cfdepends inversely on Rex
"young" BL: mdu1/dx3=t1
"old" BL: mdu1/dx3=t2<t1
Schlichting “Boundary Layer Theory” we found