1 / 18

Fluid Mechanics

Fluid Mechanics. Lecture 6 The boundary-layer equations. The need for the boundary-layer model.

jin
Télécharger la présentation

Fluid Mechanics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fluid Mechanics Lecture 6 The boundary-layer equations

  2. 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.

  3. 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.

  4. 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:

  5. 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)

  6. 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.

  7. 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.

  8. 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.

  9. x-momentum equation

  10. 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:

  11. y-momentum equation To do an order of magnitude analysis for each term and estimate the order of magnitude for

  12. y-momentum equation  Hence at most

  13. 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

  14. 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 .

  15. 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.

  16. 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)

  17. 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

  18. 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.

More Related