Clyde M. Davenport
|Introduction||Complex Solution||Elementary Properties||Conclusions|
The boundary layer equations are a pair of nonlinear partial differential equations of second order, as follows:
They are used in fluid dynamics engineering as a model of a fluid or gas flowing past a stationary boundary. Generally speaking, the fluid does not slip freely past the boundary surface. There is a flow perturbation that extends outward into the flow, the perturbed volume being known as the boundary layer. In the equations, u is the fluid velocity component parallel to the boundary, v is the velocity component in a direction perpendicular to the boundary, and α is the fluid viscosity. The perturbation is nonlinear in the perpendicular direction. Top
Below, we shall pursue a solution in terms of classical complex variables. We shall present something new in the form of the general analytical solution for the boundary layer equations in terms of complex variables. We emphasize analytical solution because the solution that we shall develop is analytic in the classical complex variable sense (i.e., is continuous and single-valued in the region of interest) and is "conservative" in physics terminology. The solution is general in that we shall first transform the pair of PDEs into one ODE and then perform two integrations to obtain a solution with two arbitrary constants of integration.
One might recall that partial differential equations, especially those used to model the behavior of some continuous material substance, typically describe the behavior of some variable, parameter, or physical effect about a point. They are typically derived from a force-energy-momentum-mass balance on an infinitesimal element at a point. If we obtain a characteristic function of the boundary layer equations by whatever means, then the function will embody the same description of physical effects as the PDE forms and must be viewed and applied in the same way; i.e., as describing the variation of effects about a point, and not necessarily the macro behavior over all space. Top
Complex Variable Solution
As mentioned earlier, our basic approach to solution is to first convert the (nonlinear) boundary layer equations to ODEs, then solve them by means of classical methods. To convert partial differentials to ordinary derivatives, we shall use the following consequences of the classical Cauchy-Riemann conditions:
where z=x+i y and f(z) is an analytic function. We are using a bold imaginary, i, because we are later going to interpret z, in specific instances, as a 2-D vector and i as a unit direction vector. These relations transform partial differentials into an ordinary derivative with respect to a 2-D independent variable, z. These relations, being direct consequences of the 2-D Cauchy-Riemann conditions, are therefore continuity conditions. By applying them to the boundary layer equations, we force continuity upon any solution(s). One might argue that we are simultaneously solving the boundary layer equations with the Cauchy-Riemann equations. However, that would be somewhat misleading because the Cauchy-Riemann conditions are satisfied by any analytic function. Top
[ASIDE: It so happens that laminar flow is analytic in the complex variable sense. Indeed, classical complex function theory has been used since the 1930s to calculate conformal flow over an airfoil shape. See Kober, H., Dictionary of Conformal Representations (Dover Publications, Inc., New York, 1957) for examples and a long list of references; also, do a Google search for "Joukowski transformation."]
Making the partial derivative conversions, we obtain:
We can rearrange these as follows:
By applying the continuity conditions (Cauchy-Riemann conditions) and converting to ODE form in this way, we are implicitly continuing both u and v into two dimensions (x,y) in such a way as to make both of them analytic functions of the complex variable, z. Integrating the first equation, we obtain: Top
where v is a classical complex constant of integration, but which can also be interpreted as the 2-D vector flow of the impinging fluid with respect to the local boundary. By the above result, it must be considered constant (but will vary in angle from point to point on a contoured boundary). Applying this relation to the second ODE, we obtain:
For reasons that shall become clear, later, we multiply both sides of the above by α and rearrange to obtain:Top
where z'=z/α. This is a simple, linear ODE which we can solve by direct integration. The result is:
where a,b,A are arbitrary complex constants of integration and v is the (constant) vector flow of the impinging fluid with respect to the tangent of the boundary surface at the point of interest. This is the general analytical solution for the u component of the boundary layer equations in terms of a classical complex variable, z. That means that it is the complete solution for the ODE form for u inasmuch as we have integrated twice and have a solution including two arbitrary constants of integration. Moreover, it is the complete solution (i.e., the most general characteristic function) of the PDE form for u, when continuity conditions are imposed. Again, we remark: Continuity conditions are nearly always assumed for PDEs, but are seldom explicitly imposed. Top
We have three arbitrary constants. We are free to set
and rearrange, to obtain the simplified form:
From the solution to the first equation, we also have an expression for v; it is:
These are the characteristic functions for the ODE forms of the boundary layer equations. Remember that v is the constant 2-D flow vector of the impinging fluid with respect to the local boundary, hence will present at differing angles on a curved surface.
We remind the reader that these do not represent the one and only, final solution of the boundary layer equations. Just as do the original PDE forms, they merely represent the behavior about a point. Given an external force, impulse, etc. on a small increment of material at the point, the characteristic functions that we have found indicate how the disturbance will be propagated away from that point in the absence of any other disturbance, reflection, interference, etc. in the medium. Top
The u,v expressions implicitly model boundary layer behavior, in the same way as does the original PDE forms. However, they do not model arbitrary scenarios in the large; instead, they represent the response of an infinitesimal element of the medium to an external impulse. If the boundary surface is curved and/or the flow is not everywhere uniform in velocity, then we must do a finite-element calculation in order to determine the shape and variation of the boundary layer. Top
However, there is a simple scenario that illustrates the main features of the boundary layer. Consider a flat, infinite boundary plate lying in a horizontal plane through the origin. Let a uniform fluid or gas flow U in the upper half space move parallel to the boundary. The constant impinging flow v reduces to the parallel component, U, only. Well away from the boundary, U is uniform and constant. There is a boundary layer, with a variable flow u(y), where y is the distance above the boundary, that drops to zero at the boundary. For some distance above the boundary (the boundary layer thickness), the flow velocity rises from zero to the the bulk flow velocity, U. We seek an explict expression for u(y) because it will show the variation within the boundary layer and can be matched to the boundary conditions.
In order to better interpret our result, here is where we wish to consider z=x+i y as a vector and shift to a different coordinate frame of reference. Recall that the imaginary i is a rotation operator (90 degrees, to be exact). If we make the linear transformation:Top
and put this into the u(z) expression, we have
Now, if we set x=0, we have
This shows the proper behavior. Recall that U is a real constant (in this particular case-there is no vertical flow), and α is the viscosity of the flowing media. We have u(0)=0, as required, then an exponential rise to the constant background, U :Top
This has all of the right behavior. There is a smooth transition from zero flow at the boundary to the uniform background flow, U. There is no clear-cut "edge" to the boundary layer, yet most of the boundary effect is confined to near the boundary. If we define the "apparent thickness of the boundary layer" as the value yb for which
then we see that, for a given fluid velocity U, the viscosity α determines the apparent thickness. Recall that U is a positive, real constant. That is, a larger viscosity produces a thicker boundary layer. Moreover, in the limit as the viscosity goes to zero (yielding inviscid flow), the boundary layer disappears and the flow slips freely past the boundary surface. [Aside: Now it is clear why we made the z'=z/α transformation in the original second order ODE; if we had not, then α would not exhibit the proper consequences.] On the other hand, for a given viscosity, α, if we increase the velocity U then the apparent boundary layer thickness is decreased. The effect is a compression, rather than a weakening, of the boundary layer.Top
The Question of Self-Similarity - The following discussion is an outgrowth of an e-mail conversation with Paul Odermalm, who pointed out that this author's solution was different from that of Ludwig Prandtl (1905) and questioned if the two could be reconciled.
Prandtl used a self-similar technique, meaning that any solution must have the same mathematical form everywhere in the boundary layer and be invariant to scaling changes on the underlying variables. This principle, applied as a starting point and a requirement, led Prandtl to the solution that he reported in 1905. The definition of a self-similar solution is one that has the form:
In the present paper, we have obtained a solution by a different method; i.e., analytic function theory. We shall prove self-similarity by recasting our solution into the self-similar standard form. Consider the following operations on our u(z) solution:
But z = x+i y, consequently: Top
Next, we merely assign:
Finally, we arrive at the self-similar form of our u(z) solution:
Similar actions apply to the v(z) solution.
We have found characteristic functions for the boundary layer equations. They can be used in place of the PDE forms in a finite-element calculation. We have shown that they have the self-similar property.
We constructed a simple formula u(y) for the boundary layer of a uniform flow past a flat plate. However, this does not generally apply for the boundary layer near a curved surface, because the impinging flow velocity, pressure, viscosity, and temperature can vary from point to point in such a regime. Again, a finite-element approach is necessary.Top