Ghosh - 550Page 112/28/2018
Velocity Boundary Layers
Prandtl first observed that the equations (A) and(B) were not useable inside the boundary layer. Although the majority of the fluid flow is without the thin boundary layer region around the body, he observed that whatever happens in this region (causing flow separation, as we shall see later) has profound effects on the flow measurements. To probe this region let us consider the steady, two-dimensional, incompressible flows:
Continuity:
x:
y:
Note: The above equations are written in the non-dimensional form with the use of the primes. We need to discuss the “order-analysis” preformed by Prandtl. It is better understood if we keep the equations in the non-dimensional form. After reaching the conclusions we’ll return to the dimensional form again.
Order Analysis by Prandtl
Prandtl hypothesized that use of the same length scale in x and y is incorrect if we need to probe inside the thin boundary layer. Thus he devised a new coordinate system in the y direction.
Ifx ~ 1(i.e. x ~ L) (Read as: “xis of the order of L”)
y ~ (i.e. y ~ L)
u ~ 1(i.e. u ~ U)
Then,
To be able to cancel the term, must be of the order 1 also.
In other words, by choosing x, y and u as the above order, the continuity equation yields the order of v as .
Here the variable is chosen to represent the size of the boundary layer and is a non-dimensional quantity (not a length). For example, if the chord of the airfoil is 1 meter, the boundary layer thickness on the airfoil (which may be a few millimeters depending on the speed of the flow) must be of order 10-2 (< 1). Typically a of ~10-3 or 10-2 means 2 is even a smaller quantity.
To perform the order analysis on the momentum equations, we note that we’ll not normally neglect the pressure terms. Since pressure is a variable which stays significant even in static fluids. However the pressure derivatives may be of different order of magnitudes. Also we have a new variable, ReL in these equations, which we must assess. We first observe that while density is a significant quantity (e.g. 1.23 kg/sec for air) viscosity is a small quantity. Thus will depend mainly on the size of (if U ~ 1 and L ~ 1). Experiments show that ~ 2 [e.g. air 1.5 x 10-5 m2/s at 20 C].
Therefore,
Based upon our observations and the result of the continuity equation we perform the order analysis of the x and y momentum equations as given below:
In the y-momentum above we notice that all the terms are order () or, smaller
[e.g., ]. Therefore the must be order (or, smaller). This is an important consideration which says although pressure is significant in the boundary layer, the pressure gradient in the direction transverse to the body is significantly small.
i.e.,.. Similarly, from the x-component we note that all the terms are order 1 or smaller. Specifically the term , which means, this term is much smaller compared to the other term. These conclusions due to the order analysis performed by Prandtl are summarized below:
(i)v < u(since, u ~ 1)
(ii)(since,
(iii)(since,
Also, since all terms in the y-momentum equation were of or smaller, Prandtl neglected the importance of this equation in his analysis.
Boundary Layer Equations
(Velocity Boundary Layers)
Based upon the foregoing analysis, Prandtl gave the following two equations in the boundary layer over any object oriented in the x-direction:
(continuity)
(x-momentum)
Note that the y-momentum is dropped out from the formulation, term is dropped out from the x-momentum and the partial derivatives on the right hand side of the x-momentum are re-written to reflect the single variable dependencies p(x) and u(y). The above two equations have 3-unknowns u, v, and p. However, note that p(x) only. This pressure solution may be known from the solution of ideal flows, solved at the edge of the boundary layer, the solution of the Euler’s equations may be computed using a CFD codes to provide the p(x) necessary at the edge of a boundary layer on arbitrarily shaped objects.
For our analysis, we will assume that is a known parameter. For a flat plate, . Boundary layer equations over a flat plate are normally solved following two approaches:
(i)Exact solution method (Blasius solution)
(ii)Approximate solution method (Karman-Pohlhausen method or, Momentum Integral Method)
Both are discussed in different sections. But before the solution methodology, let us explore further physical concepts.
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