Fluid mechanics boundary layer solution

[uby]

Hello,

I am looking for a reference which has solutions for the laminar flow boundary layer for the following scenario:

circular cylinder, L>>d, length in direction of flow, with flat circular cap
uniform laminar flow
inviscid, incompressible fluid

In other words, I would like the analog to the classical solution of an infinite flat plate (which has a parabolic dependence of the thickness of the boundary layer vs. position from the edge) for an infinite cylinder in which the flow is moving along the length of the cylinder (not around the cross-section!).

I've looked through a number of reference texts and have not found a solution.

I'm not well versed enough in the mathematics to derive my own numerical solution.

Thanks in advance!
--Dave

 

[minger]

Using the wall coordinate y=r-a; dy=dr, the axisymmetric boundary-layer equations become
$$\frac{\partial u}{\partial x} + \frac{1}{a+y} \frac{\partial}{\partial y} [(a+y)v]=0$$

$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \frac{v}{a+y}\frac{\partial}{\partial y}\left[ (a+y)\frac{\partial u}{\partial y}\right]$$
Note that if y<<a, these equations reduce to the flat plate equations. The boundary layer on a short cylinder is the Blasius solution.

Glauert and Lighthill pointed out that for $$\delta >> a$$ the convective acceleration is negligible and the momentum equation may be solved for
$$u(x,y) \approx \frac{a}{u}\tau_w (x) \ln\left(1+\frac{y}{a}\right)\,\,\,\,\delta>> a$$
That formula was then extended with the parameter $$\alpha(x) = \mu U_0 / (a\tau_w)$$ to define the profiles

For $$y \le \delta = a(e^{\alpha}-1)$$

$$u = \frac{U_0}{\alpha}\ln\left(1+\frac{y}{a}\right)$$
And for $$y\ge\delta$$
$$u=U_0$$

An algebraic expression for \alpha can be given from the momentum integral which requires some numerical technique. I can provide more information as needed, however you may be fine just using the flat plate solution.

 

[uby]

thank you for your swift reply, minger!

though the LaTeX equations did not properly show, i gather that the form of the equations has the same power dependence as the flat plate solution.

having spent some time in the library, i am sorry to report that i have only found 1 reference in the literature directly addressing this situation (LAMINAR BOUNDARY LAYER ON A YAWED
INFINITE CYLINDER ... J.C. BHATIA, Appl. Sci. Res. 30(6) 1975, p.469-75). no texts dealt with any boundary layer solutions for laminar flow other than for an infinite flat plate.

are you aware of a text the explicitly derives the formulai that you have derived here? i will need such as a reference if i were to further invoke this relationship.

(fyi, the purpose of this is so that i may model the kinetics of a gas-phase transport limited reaction while passing a reactant gas over a cylindrical substrate. my expertise is in chemistry, not fluid mechanics. my apparatus does not permit the use of anything approximating a flat plate due to proximity to the reactor walls and the drag they induce seems substantial.)

thanks!

 

[minger]

OK, I have "fixed" the latex. Apparently we're not allowed...(ooo, forgot the being equation environment before split)...I digress. Either way, for short cylinders, the flat plate solution is the cylinder solution. As the cylinder gets longer, an additional factor becomes involved.

What kind of scales are looking at?

p.s. Yes I seen the yawed cylinder too, it is for delta wing approximations.

 

[uby]

realistically, i am looking at a circular cylinder of radius 0.25 inch, length of about 0.75 inch, so a 3:2 L:d ratio with Re < 1.

 

[minger]

That would seem to indicate to me that you can safely use the Blassius solution. The long cylinder mentioned in the book references towing cables through the oceans; very long cylinders.

Go ahead and simply use flat plate, you should be fine.