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The Laminar Boundary Layer Equations
By N. Curle Dover Publications, Inc.
Copyright © 2017 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-82071-2
CHAPTER 1
INTRODUCTION
This book is about solutions of the laminar-boundary-layer equations. The concept of the boundary layer, one of the cornerstones of modern fluid dynamics, was introduced by Prandtl (1904) in an attempt to account for the sometimes considerable discrepancies between the predictions of classical inviscid incompressible fluid dynamics and the results of experimental observations. As an example, we may remark that according to inviscid theory any body moving uniformly through an unbounded homogeneous fluid will experience zero drag!
Now the classical inviscid theories assume that the viscous forces in a fluid may be neglected in comparison with the inertia forces. This, indeed, would seem a reasonable approximation, since the viscosity of many fluids (and of air in particular) is extremely small. However, in certain regions of flow, fortunately often limited, the viscous forces can still be locally important, as Prandtl observed. The reason for this is that a typical viscous stress is of magnitude µ([partial derivative]u/[partial derivative]y), where µ is the viscosity, u is the velocity measured in a direction parallel to that of the stress, and y is distance measured normal thereto, so that when the velocity gradient (or shear) [partial derivative] u/[partial derivative]y is large the viscous stress can become important even though µ itself is small. It was Prandtl who remarked that in flow past a streamlined body, the region in which viscous forces are important is often confined to a thin layer adjacent to the body, and to a thin wake behind it. This thin layer is referred to as the boundary layer. When this condition holds the equations governing the motion of the fluid within the boundary layer take a form considerably simpler than the full viscous-flow equations, though less simple than the inviscid equations, and it is the solution of these equations with which we shall be presently concerned.
An alternative method of looking at this concept is as follows. The inviscid-flow equations are of lower order than the viscous-flow equations, so fewer boundary conditions can be satisfied in a mathematical solution of a given problem. Thus an inviscid-flow solution allows a finite velocity of slip at a solid boundary, whereas the solution of the viscous-flow equations does not allow such slip. In other words, the inviscid-flow solution assumes the existence of an appropriate vortex sheet at the solid boundaries. Now in reality this vorticity will diffuse outwards from the boundary (in much the same way that heat would diffuse from a heated body) and will be convected with the stream. Thus, considering now the flow past a flat plate, the time t in which fluid travels a distance x parallel to the plate will be of order x/U, where U is a typical velocity, and in this time the vorticity will have diffused outwards through a distance of order (vt)1/2 (vx/U)1/2, where v = µ/ρ is the kinematic viscosity. This is an indication of the boundary-layer thickness.
Before turning to a more quantitative discussion, mention must be made of the phenomenon of boundary-layer separation. When the fluid is proceeding into a region of rising pressure, it is slowed down by this retarding force. In the outer part of the boundary layer, where the kinetic energy is large, this results only in a relatively small slowing down of the fluid, but the effect on the slower-moving fluid nearer to the wall can be considerable, and if the pressure rise is sufficient it can be brought to rest, and, farther downstream, a slow back-flow be set up. In such circumstances the forward flow must leave the surface to by-pass this region, and boundary-layer separation is said to have taken place. If the region of separated flow is extensive, the separation can have a back-reaction on the external flow, which is then quite different from what it would have been in the absence of the boundary layer. If the separated region is limited, on the other hand, the external flow may not be significantly affected, and the flow field may be calculated by calculating firstly the external flow (on the assumption of no boundary layer) and then calculating the boundary layer appropriate to this external flow.
The above qualitative analysis is restricted to low-speed flows, but serves to indicate the nature of the boundary layer in a simple way. At supersonic speeds, for example, interactions of the boundary layer with the external stream become more important, and lead to considerable theoretical difficulties which will not be discussed at this stage. In what follows a quantitative analysis will be given of how the boundary-layer equations may be deduced from the exact equations of viscous flow, and detailed discussions will be given of some of the points touched upon briefly above.
1. The equations of viscous flow
We take cartesian coordinates (x,y), with associated components of fluid velocity (u, v). The fluid is assumed to have pressure p, density ρ, and absolute temperature T, and these are functions of x and y only, in view of the approximation (made throughout this book) that the flow is two-dimensional and steady. The equations of motion then express the basic physical ideas that for a given element of fluid there is conservation of mass, momentum (excepting in so far as the element is acted upon by various forces), and energy (excepting in so far as work is done by these same forces). For a general derivation of these equations, reference may be made to volume I of Modern Developments in Fluid Dynamics, High Speed Flow (Howarth, 1953). In steady two-dimensional flow the equation of conservation of mass takes the form
(1) [MATHEMATICAL EXPRESSION OMITTED]
This equation is usually referred to simply as the equation of continuity. The equations of conservation of momentum in the x and y directions (the momentum equations) become
(2) [MATHEMATICAL EXPRESSION OMITTED]
and
(3) [MATHEMATICAL EXPRESSION OMITTED]
where
(4) [MATHEMATICAL EXPRESSION OMITTED]
and X,Y are the external forces per unit mass of fluid. Finally the equation of conservation of energy (the thermal energy equation) takes the form
(5) [MATHEMATICAL EXPRESSION OMITTED]
where
(6) [MATHEMATICAL EXPRESSION OMITTED]
There are thus four equations, (1), (2), (3), and (5), for the five unknowns u, v, p, ρ T, and these are soluble, in principle at any rate, when an equation of state is defined, relating p, ρ and T. For a perfect gas this takes the form
(7) p = ρ RT,
where R is the gas constant.
In the above equations cp is the specific heat at constant pressure, usually taken to be constant, and k is the thermal conductivity, related to the thermometric conductivity κ by the relationship
κ = k/ρcp.
For a physical interpretation of the quantities κ k, µ and v = µ/ρ the reader is referred to the book Modern Developments in Fluid Dynamics (Goldstein, 1938). It will suffice for the present to remark that v is a parameter determining the rate at which vorticity is diffused, whilst κ determines the rate at which heat is diffused. The ratio
σ = v/κ = µcp/k
accordingly determines the relative rates of these two types of diffusion, and is called the Prandtl number. It is usual to assume that the Prandtl number is constant, and this holds true for air over quite a wide range of conditions, the value being about 0·72. It will be seen later that considerable simplifications are often possible if it is assumed that the Prandtl number is unity, an approximation that is not without value for air.
2. Boundary layers
It will be noted that the viscosity µ appears in equations (2) and (3) only as a multiplicative factor of velocity gradients, or of powers or products of velocity gradients. Accordingly, if the viscosity is small, classical fluid dynamics theory, which neglects viscosity, will be valid except in regions where velocity gradients are large. Similarly, provided the Prandtl number s is not too small, small viscosity implies small thermal conductivity, so that the terms involving k in (5) are important only where the temperature gradients are large.
Now it is often found in practice that the regions of high velocity and temperature gradients are confined to a narrow region near to solid walls, known as the boundary layer, and to a thin wake behind streamlined bodies. In such a domain, considerable simplifications of the equations of motion are possible, even though all the terms involvingµ and k cannot be neglected, as was first shown by Prandtl (1904) in a paper of fundamental importance. In low-speed flow it is usually possible to consider the development of the boundary layer as a separate problem from that of the substantially inviscid external flow, the exception being when there is a substantial region of separated flow. When the external flow is supersonic, however, there is an interaction between the boundary layer and the external stream which must be taken into account. Crudely we may say that though an artificially introduced disturbance cannot be propagated upstream in a wholly supersonic flow, the presence of a boundary layer, in which the flow sufficiently near to the walls will be subsonic, provides a mechanism for such upstream influence. Accordingly the external stream does not approximate to that obtained in the absence of the boundary layer, and cannot be independently prescribed.
In spite of this difficulty it is still useful to begin by considering separately the boundary-layer approximation and the inviscid approximation, as use can be made of these results even in certain problems in supersonic flow where the interaction between boundary layer and external stream is particularly important, as, for example, when a shock wave interacts with a boundary layer.
3. The laminar boundary layer on a plane wall
We begin by deriving the boundary-layer equations for flow over a plane wall. The x-axis is taken along the wall and the y-axis perpendicular to it. We assume that the thickness δv of the layer in which the velocity gradient [partial derivative]u/[partial derivative]y is large, where the velocity u rises rapidly from 0 at the wall to a value u1 is much smaller than a typical length l in the flow field as a whole. Equally well we assume that δt the thickness of the layer in which the temperature gradient [partial derivative]T/ [partial derivative]y is large, and in which the temperature of the fluid changes from the temperature of the wall Tw to a value T1 is also much less than l. We shall assume at this stage that δv and δt are the same order of magnitude, each being of order δ << l. This restriction can later be removed, provided δv and δt are both << l. We let u0 ρ0T0 be typical values of velocity u, density ρ temperature T, and may then deduce the order of v from (1). This equation shows that
[MATHEMATICAL EXPRESSION OMITTED]
which is of order pu/l. Hence, upon integrating across the layer from y = 0 to y = δ we find that ρv is of order (ρ0u0/l)δ so that
[MATHEMATICAL EXPRESSION OMITTED]
Thus v is small compared with u, within and at the edge of the boundary layer.
We can now examine the magnitudes of the various terms in (2). In doing so we remember that the derivative [partial derivative]F/[partial derivative]x of a function F will be much smaller than the derivative [partial derivative]F/[partial derivative]y. In fact
[MATHEMATICAL EXPRESSION OMITTED]
Thus
(8) [MATHEMATICAL EXPRESSION OMITTED]
(9) [MATHEMATICAL EXPRESSION OMITTED]
and
(10) [MATHEMATICAL EXPRESSION OMITTED]
Ideal fluid-dynamics theory rejects all the terms involving viscosity, namely those in (9) and(10), but boundary-layer theory retains the former, since δ2 is small as well as µ. We see that these terms are the same order of magnitude as (8) when
(11) [MATHEMATICAL EXPRESSION OMITTED]
where R is the Reynolds number
[MATHEMATICAL EXPRESSION OMITTED]
We have chosen δv as the relevant value of δ since we are considering a momentum equation. Upon rejecting the terms in (10), then (2) simplifies to
(12) [MATHEMATICAL EXPRESSION OMITTED]
We note that the rejected terms are of order (δ/l) times those retained. We now deal in a similar manner with equation (3). We can see that
(13) [MATHEMATICAL EXPRESSION OMITTED]
(14) [MATHEMATICAL EXPRESSION OMITTED]
and
(15) [MATHEMATICAL EXPRESSION OMITTED]
Accordingly, the terms in (15) are of order δ/l times those rejected in (2), and those in (13)and (14) are of order δ/l times those retained. It follows that
(16) [MATHEMATICAL EXPRESSION OMITTED]
Now Y will usually be zero. For example body forces can be neglected in problems of forced convection, and in problems of free convection the body force (gravity) will act in the x direction. Assuming, then, that Y = 0, it follows from (16) that the pressure gradient [partial derivative]p/[partial derivative]y is small, and the pressure change across the boundary layer is very small, being [MATHEMATICAL EXPRESSION OMITTED], which is neglected. Thus (3) reduces simply to
(17) p(x,y) = p(x).
We turn now to the thermal energy equation (5). By identical reasoning to that given above it follows that the two terms on the left-hand side are of equal order of magnitude. The term [partial derivative]([partial derivative]p/[partial derivative]y) vanishes by (17), and the term [partial derivative](k[partial derivative]T/[partial derivative]x)/[partial derivative]x is of order (δ/l)2 times [partial derivative](k[partial derivative]T/[partial derivative]y)/[partial derivative]y. Finally, in Φ, the term ([partial derivative]u/[partial derivative]y) is [MATHEMATICAL EXPRESSION OMITTED], which is at least (l/δ)2 times the other terms; these are therefore neglected. It follows that (5) becomes
(18) [MATHEMATICAL EXPRESSION OMITTED]
When body forces are negligible, an alternative form of (18) is obtained by adding u times(12) to (18). This yields
(19) [MATHEMATICAL EXPRESSION OMITTED]
(Continues...)
Excerpted from The Laminar Boundary Layer Equations by N. Curle. Copyright © 2017 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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