26-01-2015, 07:49 PM
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car boundary layer theory ppt
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23-03-2017, 04:04 PM
Theory of the boundary layer
We start from Newton's Second Law of Motion and we obtain the equation that describes the conservation of momentum for the fluid passing through a control volume. We have derived this first integrally. To obtain detailed relationships between pressure, velocity and shear forces at each point, we need the differential form of the momentum equation. To obtain this, we must first bring the integral equation under a single type of integral, so that we can establish the integrand at zero. In turn, this is achieved by converting surface integrals to volume integrals and vice versa, using the appropriate vector analysis theorems: the Divergence Theorem in this case. (Note: a similar transformation between line integrals and surface integrals can be obtained using Stokes' theorem).
The differential form of the momentum equation basically says:
"Constant rate of momentum change per unit volume, at one point + convective momentum rate per unit volume = pressure force + shear force + body force, per unit volume."
If the inertial forces are much greater than the forces of the body, we can neglect the forces of the body. Examples in which we can not neglect the forces of the body are: fluid flow (the gravitational force must be considered in general); Or ion flux in a magnetic field (electromagnetic forces must be considered).
If the forces of inertia are much greater than the shear forces, we can also neglect the terms of shear force. This occurs at a high number of Reynolds, in out-of-surface flows.
If shear forces are neglected (regardless of whether body forces are neglected or not), the resulting equation is called the Euler equation. The Euler equation is used in many sophisticated calculations of aerodynamics, rotary machines, and compressible flow over the wings.
If we continue to apply "irrotationality" conditions, we can rewrite the momentum equation, combined with the mass conservation equation, as a "potential equation", as we have seen earlier in low-speed aerodynamics, and compressible aerodynamics.
Let's go back to the situation where we can not neglect the shear forces: it flows near the limits of the fluid. These are generally thin layers, which we will call "boundary layers" (what else ?? :-) In these regions, we must retain the full momentum equation including the shear force terms
We must relate the shear stress to the velocity, density, etc., which are the flow variables of interest. In the case of solids, shear stress is related to shear deformation. If the relation is linear, the proportionality constant is called the shear modulus.
In the case of fluids, there is no resistance to mere shear stress: fluids can not resist changing shape. However, there is resistance to the rate of deformation. In the ideal case of a Newtonian fluid (not to be confused with Newtonian flow), the relationship is linear and the proportionality constant is called "absolute viscosity", a property of the fluid in question. And the example of a "non-Newtonian" fluid is blood, where the resistance is much lower if the rate of deformation is applied parallel to the platelets, perpendicular to them.
We assume that the fluid is Newtonian. Next, we also assume that the "mass viscosity" is zero. Bulk viscosity is the proportionality constant between normal stress (such as pressure) and normal strain rate. In the case of very rapid compression, such as that of the leading edge of a shock wave, the bulk viscosity may be significant and is a cause of loss of stall pressure through a collision. In most other situations, the normal deformation rate is too small to be concerned with bulk viscosity.
This is called the Stokes hypothesis.
The momentum equation with this applied hypothesis is called the Navier-Stokes equation. It is highly nonlinear. We need to look for simplified forms to use in the boundary layer. To do this, we use some facts that we observe about the boundary layers:
1. The space velocity at which properties change across a boundary layer is very large, compared to the rate at which things change along the direction of the flow. In other words, derivatives with respect to and are much higher than derivatives with respect to x.
2. The static pressure is constant across a boundary layer.
These two facts are used to simplify the Navier-Stokes equations in a set of "layer equations".
The whole character of the equation changes from the elliptic nature of the N-S equns to the parabolic nature of the equations B-L. It is possible to march downstream and resolve a boundary layer.
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