VIII. Derivation of the Boundary Layer Equations

	The Navier-Stokes equations that we studied in the previous three chapters were developed around 1845. These equations were of limited use to the practicing engineer because of the following reasons:

1.	These equations are highly nonlinear. As a result, engineers could not use classical approaches for solving PDEs such as superposition of several simple solutions (similar to the superposition of sources, sinks, vortices in potential flow) , and the separation of variables approach where the flow properties (u,v,p,r) where each assumed to be a product of simple functions F(x) and G(y).

2.	The steady state (∂/∂t=0) form of these equations are elliptic PDEs. This simply means that the flow properties at a point in a flow field are influenced by every other point in the entire flow field. For example, according to Navier-Stokes equations, the skin friction at a point on the nose of a jumbo jet will depend on the behavior of a point way downstream on the tail. This necessity to deal with the entire flow field where all the points are coupled to each other lead to formidable mathematical and numerical difficulties during the turn of the 20th century. 

	Engineers at that time (and now) were interested in the following information:

	a) 	For a given configuration, where does separation occur?

	b)	For a given configuration, where does the flow transition from a laminar flow with low skin friction drag to turbulent flow with high skin friction?

	c)	For a given configuration, what is the variation of skin friction and boundary layer thickness along the surface?

	Prandtl developed the boundary layer theory around 1904. As an engineer, he was willing to simplify the governing equations by dropping certain terms in the Navier-Stokes equations, provided these terms could be shown to be significantly smaller than the terms retained. He used an order of magnitude analysis for this purpose. To illustrate how this will work, let us first use an order of magnitude analysis based on a single reference velocity V and a single length scale L.

Order of Magnitude Analysis based on a Single Velocity Scale and a Single Length Scale:

For the sake of simplicity, we will consider in detail  steady, incompressible viscous flows, and write down the compressible form later. Consider the continuity equation first.

	An order of analysis says that the velocity fields u and v appearing in the above equation are of order O(V), while the distances x and y are of order O(L). Thus, both the terms in the continuity are of order O(V/L). Neither term can be dropped.

	Consider the u- momentum equation next. 

	Substituting V where ever we see a 'u' or a 'v' , and substituting L where see an 'x' or 'y' , we can carry out an order of magnitude of analysis.  The left hand side terms (called inertial terms) are of order O(V2/L). The terms on the right hand side of the above equation are of order O(mV/L2). The ratio of these two group of terms (inertial and viscous) is of order O(rVL/m) , or of order O(Re) were Re is the Reynolds number.  For high Reynolds number flows, our order of magnitude analysis says that the u- momentum equation may be approximated as 

	The above equation is simply the inviscid form of the Navier-Stokes equations.

	By an identical order of magnitude analysis, we can show that the v- momentum equation

	Thus, our order of magnitude analysis based on a single velocity scale and a single length scale gives us the inviscid flow equations (also known as the Euler equations), derived in AE 3003. While these equations have several merits, and form the basis of the potential flow theory, they can not explain why 2-D airfoils experience drag, while potential flow says 2-D airfoils can not experience drag. This conflict between the inviscid flow theory and experience is known as d'Alembert's paradox.

Prandtl's Order of Magnitude Analysis based on Multiple Length Scales:

	Prandtl's analysis was guided by intuition and experimental observations. He found that the viscous flow over most surfaces (external or internal) may be divided into two regions - an external region where the appropriate length and velocity scales are V and L, and an internal region closed to the solid surface, called a boundary layer where flow properties vary rapidly in the direction normal to the solid surface. In this outer region, our earlier order of magnitude of analysis is acceptable, and the flow equations reduce to inviscid flow equations. In the inner region, across a small distance (less than 10 millimeters for an airfoil that is one meter long) the velocity may vary .rapidly from 0 to freestream velocity. In the streamwise direction, the variations are considerably small. Prandtl proposed using a length scale d while assessing the gradients in the normal y- direction, across the boundary layer. This length scale may be viewed as a rough measure of the boundary layer thickness, say at midchord. In the streamwise direction the length scale was, as before, L.

	As far as velocities go, the u- component of velocity within the boundary layer is found from a large number of measurements to be of the same order as the freestream velocity V. Thus, we can normalize u by V. The v- component of velocity was, as a rule, considerably smaller than u or V. Rather than picking a velocity scale for v, Prandtl decided to let his order of magnitude tell him how v should be normalized.

	Let us assess the magnitude of all the terms in the Navier-Stokes equations, using the two length scales d and L , and the velocity scale V for u that Prandtl chose.

Continuity:

		Term 1: ∂u/∂x		Order of Magnitude:  	O(V/L)

		Term 2: ∂v/∂y	Order of Magnitude:	O(v/d)

	Obviously, these two terms must be of the same overall order of magnitude. otherwise, they will never balance with each other and yield a sum of zero. Therefore, Prandtl concluded that the vertical component of velocity v is of order

u- Momentum Equation:

	Let us next consider the u- momentum equation:

	The order of magnitude of the individual terms are as follows:

	We put a question mark for the pressure gradient, because we do not know how pressure behaves within the boundary layer, at this point. We do know that in the outer region, pressure is of same order as rV2, from Bernoulli's equation, but we do not want to assume a similar order of magnitude for p within the boundary layer, yet.

	Consider the first viscous term on the right side. Clearly, since L is much larger than d, the term m ∂2u/∂x2  is much smaller than m ∂2u/∂y2.. Therefore, as we start dropping terms, this term m ∂2u/∂x2 will be the first one to go.

	How about the second term  m ∂2u/∂y2 ?  Since m is a small number, we are sorely tempted to drop this term (reducing our equations to inviscid equations), but realize that we have a small quantity d2 in the denominator. For this term to make any significant impact, it must be of the same order as the rest of the inviscid terms. That is,

	If the above statement is true, then it follows that d must be of the following magnitude:

Thus,

(1)
	The above equation is very powerful, despite its simple appearance. It states that the thickness of a boundary layer is inversely proportional to the square root of the Reynolds number. The higher the Reynolds number, the thinner the boundary layer is likely to be.

	Using what we learnt about the order of magnitude of v, we can also write

(2)
	This relationship says that in high Reynolds number flows, the v- component of velocity is inversely proportional to the square root of the Reynolds number.

Caution: Our order of magnitude estimates strictly hold only for laminar flows, where the mixing and viscous effects are due to molecular processes. In turbulent flows, we can not use them directly, but may still be guided by them.

	We can now close the book on the term 1/r ∂p/∂x. This term is at most of the same order, as the rest of the terms, i.e.  O(V2/L). It can, of course, be smaller. Since we do not know if this term is going to be small, we do not drop this, but retain it. 

	In summary, an approximate form of the u- momentum equation based on Prandtl's order of magnitude equation is then:

(3)
v- Momentum Equation:

	We finally turn our attention to the v- momentum equation.  Again, we leave the term ∂p/∂y alone, since we do not know its order of magnitude, a priori.

	From the above order of magnitude of analysis, (where we have used d/L behaves inversely with the square root of Reynolds number to remove d) we find all the identified terms are small. Most are of order , while the term m∂2v/∂x2 is even smaller, of order . In high Reynolds number flows, all these terms may be neglected. Since ∂p/∂y appears in this equation, this term can not be very large. otherwise, things will not balance. It can at most be of order  . The v- momentum equation, under Prandtl's order of magnitude then takes on the following simple form:

(4)
	The above equation implies that pressure does not vary significantly across the thickness of the boundary layer. One can, therefore,  compute the pressure at the edge of the boundary layer, from inviscid flow theory. In other words, pressure field is computed from a knowledge of the outer inviscid flow region, and not computed as part of the boundary layer calculations.

	In summary, the 2-D incompressible, boundary layer equations are:

(5)
	Notice that we have begun to use the ordinary derivative dp/dx, rather than the partial derivative ∂p/∂x. This is because ∂p/∂y = 0 from boundary layer analysis, and p within the boundary layer is only a function of x.

Boundary Conditions:

	Equation set (5) must be complemented by an appropriate set of boundary conditions. Consider the boundary layer shown below:

Solid Wall:

	At the solid wall, no slip conditions are employed. This requires u=0, v=0 at stationary walls. If the wall is moving, or if there is mass blown into the boundary layer, then the user/designer must specify the surface velocity u, and the blowing velocity v.

Boundary layer Edge:

	The PDE governing the behavior of u is second order in y. This means we can specify two boundary conditions regarding the behavior of u(x,y). The most logical boundary condition is to require u at the boundary layer to match u(x,y) with the external, inviscid flow field ue(x).

	The PDE governing the v- velocity field is, from equation set (5), just the continuity equation. This is first order in y. therefore, only one boundary condition (either at the solid surface, or at the boundary layer edge) may be prescribed. The usual practice is to impose no slip boundary condition v=0 at the solid surface, and let the value of v at the boundary layer edge be determined from the continuity equation. A mismatch between the v- component of velocity from the external potential flow analysis, and the v- component from the boundary layer analysis is inevitable. This mismatch may, however, be removed, if we follow an iterative procedure, where the external inviscid flow is recomputed using the v- velocity at the edge of the boundary layer as the boundary condition. Such an iterative procedure is known as viscous-inviscid interaction procedure, and is used in some airfoil analyses used in industry.

Upstream and Downstream Boundaries:

	An examination of the governing equations shows that the highest derivative in the x- direction is of first order. Our boundary layer equations, therefore, require only one boundary condition. The correct boundary condition is to specify u(x,y) at an upstream location (Nose of the airfoil, Upstream end of a duct, etc.)  We can then march one x-station at a time, and stop whereever we want. There is no need to march all the way to the end of the physical domain (airfoil trailing edge, end of a diffuser duct) unless we want to.

	Recall the difficulties cited with the Navier-Stokes equations because they were 'elliptic'. The boundary layer equations are 'parabolic'. This means the flow behavior at a particular point depends only on the information at that station, and information upstream. The downstream velocity field is not required to compute u(x,y) or v(x,y).

Justification for the Use of Cartesian Form of Equations over Curved Surfaces:

	One question that may puzzle the beginner is the use of a Cartesian equation to formulate the boundary layer equations. After all, the airfoil surface is curved. Shouldn't we be using a curvilinear coordinate system, that accounts for, among other things, the curvature of the airfoil surface?

	The answer to this question is as follows. In our development, the x- axis is along the airfoil surface, and y- axis is normal to the airfoil surface. We use a flat earth analogy, and assume that the boundary layer thickness d is very small compared to the airfoil surface radius of curvature. Then, just as in architecture where we usually neglect earth's curvature while designing buildings and use a Cartesian coordinate system, we can neglect the airfoil surface curvature, and assume that the surface is flat.

Second Order Boundary Layer Theory: Prandtl's boundary layer theory, and equations (5) are sometimes known as the first order boundary layer theory, because we neglected terms of order O(d). We also neglected airfoil curvature, saying that d is much smaller than the surface radius of curvature. There are higher order theories available, where additional terms are kept and only terms of order d2 are neglected. Such a theory, which will account for surface curvature effects, is known as a second order boundary layer theory. Since first order boundary layer theory gives such good results, second order boundary layer theories are seldom pursued, except in the halls of academia.

Compressible Form of the Boundary layer equations:

	The compressible form of the boundary layer equations may be derived by a very similar order of magnitude analysis. While the derivation is long, it is not particularly difficult. The final form of the boundary layer equations are given below:

Singularity of Boundary layer Equations at Separation Point

	Consider the development of the boundary layer over a solid surface. In some parts of the flow, where favorable pressure graident exists (∂p/∂x < 0) , the flow may be along the positive x- direction everywhere. Such a situation is called an attached flow.

	If the pressure gradient chages from favorable to adverse at a point downstream, then the combined effects of the viscous friction, and the pressure gradient may slow down the fluid significantly. A point is reached, where both u and its derivative ∂u/∂y become zero. This point at the solid surface, where u and ∂u/∂y simultaneously vanish (i.e. go to zero) is known as separation point.
	Downstream of this point, part of the flow is moving with a negative u- component of velocity. This region, where the flow is reversed is called the reversed flow or separated flow region.
	Recall that our strategy for solving the boundary layer equations require marching in the x- direction, from some upstream point where the flow properties are known. In effect, we are solving

	In the vicinity of the separation point, at several points in the flow field u becomes uncomfortably close to zero. At the separation point, there is at least one point just above the wall where u is exactly zero. This means the right hand side of the above equation becomes "singular" or undefined at the separation point. Our marching process thus comes to a grinding halt at the separation point. Downstream of the separation point, some other means of solving the boundary layer equations must be pursued. One approach is to prescribe the shear stress at the wall, and solve for the edge velocity ue that will produce this skin friction. This is the inverse of the standard approach where ue is prescribed at the boundary layer, and we solve for u and skin friction at the wall. Such inverse methods are beyond the scope of the present study.

Note: There are more formal methods of showing that the boundary layer equations become singular at the separation point.

Summary:
The 2-D steady, incompressible laminar boundary layer equations are:

	The 2-D steady, compressible boundary layer equations are:

Some observations are:

a)	The variation of pressure across the boundary layer is small. Thus, we can assume that pressure varies only with respect to x. The pressure field may, therefore, be computed at the edge of the boundary layer from inviscid flow theories, without any knowledge of the boundary layer characteristics.
b)	The v- component of velocity within the boundary layer is of the order of O(Vd/L) where d is the boundary layer thickness.
c)	The boundary layer thickness d is very small, and for laminar flows it varies inversely as the square root of Reynolds number rVL/m.
d)	We will assume the airfoil surface to be flat, and use a Cartesian coordinate system, neglecting surface curvature. This is because the boundary layer thickness d is very small compared to the airfoil surface radius of curvature.