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Fluid Mechanics. Lecture 6 The boundary-layer equations. The need for the boundary-layer model.
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Fluid Mechanics Lecture 6 The boundary-layer equations
The need for the boundary-layer model • While the flow past a streamlined body may be well described by the inviscid flow (and even the potential flow) equations over almost all the flow region, those equations do not satisfy the fact that – because of finite viscosity of real fluids – the flow velocity at the wall itself must vanish. • So, we need a flow model that uses the simplest possible form of the Navier Stokes equation but which does enable the no-slip condition to be satisfied. • Such a model was first developed by Ludwig Prandtl in 1904.
Objectives of this lecture • To explore the simplification of the Navier Stokes equations to obtain the boundary layer equations for steady 2D laminar flow. • To understand the assumptions used in deriving these equations. • To understand the conditions in which the boundary-layer equations can be used reliably.
The governing equations • Navier-Stokes equations: • We seek to simplify these equations by neglecting terms which are less important under particular circumstances. • Key assumptions: the thickness of the region where viscous effects are significant,δ, is very thin , i.e. d << L and ReL >>1. Continuity: x-momentum: y-momentum:
Non-dimensionalized form of N-S Equations L L • Non-dimensional-ize equations using V, a constant (approach) velocity, L ,an overall dimension i.e. • U*= U/ V; V*=V/V; x*=x/L; y*=y/L • P*=??? ( A question for you)
Non-dimensionalized N-S equations L L x has a magnitude comparable to L • Since , . • Hence we write x* has an order of magnitude of 1.
Non-dimensionalized N-S equations L L • Since and , . • Hence we writey*= O(d). • Also we have y* is at least an order of magnitude smaller than 1.
Continuity equation [V*] has to be of order O(d) to satisfy continuity, i.e.. No term can be omitted hence the continuity equation remains as it is, i.e.
x-momentum equation To make the above equation valid, we must have: ReL has to be large and x-gradients in the viscous term can be dropped in comparison with y-gradients. The dimensional form of the equation thus becomes:
y-momentum equation To do an order of magnitude analysis for each term and estimate the order of magnitude for
y-momentum equation Hence at most
y-momentum equation L L The pressure can be assumed to be constant across the boundary layer over a flat plate. Hence the pressure only varies in the x-direction and the pressure at the wall is equal to that at the edge of the layer, i.e. U
Two qualifiers • If the surface has substantial longitudinal curvature (/R >0.1) it may not be adequate to assume constant pressure across boundary layer. Then one needs to apply radial equilibrium to compute P (see Slide 16) • In 3D boundary layers (not covered in this course but very important in the industrial world) one needs to be able to work out the presssure variations in the y-z plane (normal to the mean flow) to compute the secondary velocities .
Summary of assumptions • Basic assumption: • Derived results • V is small, i.e. • Re must be large: and then only velocity gradients normal to the wall are significant in the viscous term • The pressure is constant across the boundary layer (for 2D nearly straight) flows, i.e.
Boundary layer equations L Continuity x-momentum • Since disappears, the equations become of parabolic type which can be solved by knowing only the inlet and boundary conditions... i.e. no feedback from downstream back upstream. • Unknowns: U and V; (P may be assumed known) • Boundary conditions: At wall : Free stream: Inlet: L y = 0; U = V = 0 y = d; U = V U U(x0,y), V(x0,y)
Boundary layer over a curved surface Pressure gradient across boundary layer: Assume a linear velocity distribution, i.e. integrating from y=0 to d gives Hence pressure variations across the boundary layer are negligible when
Limitations • Large Reynolds number, typically Re >1000 • Boundary-layer approximations inaccurate beyond the point of separation. • The flow becomes turbulent when Re > 500,000. In that case the averaged equations may be describable by an adapted for of momentum equation – to be treated later. • Applies to boundary layers over surfaces with large radius of curvature.