Claculation of Monentum thickness on boundry layer?

[GuestR01312011]

Hi, Im struggling with a problem that requires me to find the position from the leading edge whereupon the flow translates from Laminar to Turbulent, using the Von Karmann integral eqtn/s. Anyone know if this is possible?
Oh yeah I dont know the free stream speed or the boundry layer speeds. Also can I calculate the momentum and Displacement thickness' not in terms of the boundry layer thickness without knowing the velocities of the profiles??? I have the local Reynolds number at the transition point but dont have the boundry layer vel.?
Thanks a million!
Its for a flat plate by the way.

[jehardiman]

Look up Thwaites method. "Approximate calculation of the laminar boundary layer" Aeronautical Quarterly I, 1949.

But if you have the Rn at transition why do you need anything else if all you are looking is the location?

[GuestR01312011]

I need the location to find the stream speed and hence the momentum & displacement thickness.

[jehardiman]

If you have the Rn, then you know the relationship between U and x. And as x will vary inversely with U, and you stated that don't have U, the problem is sloved ...so what's the issue you are looking for? Because you don't have enough info for a discreet answer without iteration.

[tspeer]

...find the position from the leading edge whereupon the flow translates from Laminar to Turbulent, using the Von Karmann integral eqtn/s. ...
I dont know the free stream speed or the boundry layer speeds.

If you don't know the free stream speed, the problem can't be solved. The boundary layer development is totally dependent on the free stream velocity profile.

Also can I calculate the momentum and Displacement thickness' not in terms of the boundry layer thickness without knowing the velocities of the profiles???

The method of VonKarman and Pohlhausen basically assumes a family of shapes for the velocity profile that is defined by the shape factors Lambda and Kappa.
Lambda = delta^2/nu * dU/dx
Kappa = delta2^2/nu * dU/dx = Z * dU/dx
delta = boundary layer thickness
delta2 = momentum thickness
U = velocity outside the boundary layer
dU/dx = velocity gradient outside the boundary layer

The form of the velocity profile they chose is
u/U = (2 eta - 2 eta^3 + eta^4) + Lambda/6 * (eta - 3 eta^2 + 3 eta^3 - eta^4)
eta = y / delta(x)
delta(x) = local boundary layer thickness

Kappa and Lambda are related by
Kappa = (37/315 - 1/945 Lambda - 1/9072 Lambda)^2 * Lambda

The ratio between the displacement thickness and the momentum thickness, H12, is given by
H12 = delta1/delta2 = (3/10 - 1/120 Lambda) / (37/315 - 1/945 Lambda - 1/9072 Lambda^2)

Once you know Lambda, you have the whole velocity profile.

I have the local Reynolds number at the transition point but dont have the boundry layer vel.?...Its for a flat plate by the way.

The way you've posed the problem doesn't make much sense. If it's the Reynolds number based on length and a free stream velocity, then you have the distance from the leading edge from the definition of the Reynolds number. If the Reynolds number is based on, say, the local momentum thickness and local outside velocity, then you'll have to calculate the development of the boundary layer - and for that you need the whole outside velocity profile from the stagnation point.

The vonKarman integral equations describe the development of the laminar boundary layer. In themselves, they say nothing to do with transition. You need to apply stability theory or some transition criterion to find out when transition occurs. But you say you already know the transition Reynolds number.

With a whole lot of substituting of variables, the momentum equation for the boundary layer reduces down to
dZ/dx = F(Kappa)/U
Kappa = Z * U'
F(Kappa) = 2(37/315 - 1/945 Lambda - 1/9072 Lambda^2)*[2 - 116/315 * Lambda + (2/945+1/120)*Lambda^2 + 2/9072 Lambda^3]
Lambda = 7.052 at the stagnation point
Z = 0.0770 / U' at the stagnation point
dZ/dx = -0.0652 * U''/U'^2 at the stagnation point

To calculate the development of the laminar boundary layer, perform the following steps:
1. The velocity outside the boundary layer, U(x) and it's derivative dU/dx must be known as a function of the arc length, x.

2. Integrate the momentum equation to get Z and the shape factor Kappa. Kappa gives you the momentum thickness.

3. Get Lambda from Kappa using the relationship above.

4. Once you have Lambda, you have the velocity profile. You can get laminar separation point from this, or the local amplification factor for a stability-based transition criterion.

For a flat plate in a uniform stream, U(x) = free stream Uo, U'= 0. This means dZ/dx = 0.4698 * Uo and delta2 = 0.686*sqrt(nu*x/Uo).

See Schlichting, Hermann, "Boundary Layer Theory", McGraw-Hill, 1979. Chapter X.

[GuestR01312011]

Thanks guys, just found out we have to assume either a transition length or free stream speed to continue.