Momentum Heat Mass TransferMHMT6Solutions of the Navier-Stokes equations in limiting cases. Engineering Bernoulli equation. Boundary layer theory. Flow along a plate. Karman integral theorem.Rudolf itn, stav procesn a zpracovatelsk techniky VUT FS 2010Bernoulli equation. Boundary layer.
Navier Stokes - high ReMHMT6Let us consider the 2D Navier Stokes equation describing for example a steady state fluid flow around a body (diameter D) at incoming velocity U. The following equation represents a momentum balance in the incoming flow direction and should have been used for prediction of drag forces Introducing dimensionless variables X=x/D, Y=y/D, Ux=ux/U, Uy=uy/U, Re=UD/ gives dimensionless equationIt seems to be obvious that with the increasing velocity (with increasing Re) the Navier Stokes equation reduces to the Eulers equation and the last term of viscous forces is less and less important (all dimensionless variables X,Y,Ux,Uy are supposed to be of the unity order, with the exception of Re).
dAlemberts paradoxMHMT6DAlemberts paradox. Analytical solutions based upon Euler equation indicate, that the resulting force (integrated along the whole surface of body) should be zero. It was a great challenge for the best brains of 19th century (Lord Rayleigh, Lord Kelvin, von Karman) to explain the controversy between experience (scientists knew about the quadratic increase of drag forces with velocity U) and theory represented either by the Stokes solution for drag on a sphere (cD=24/Re) or steady state solutions of Eulers equations predicting zero drag for high Reynolds number. Suspicion was focused to possible discontinuities/ instabilities of potential flows, and to wake, resulting to explanation of many important phenomena, for example the Karman vortex street (see previous lecture, or read von Karman paper, T.Karman: ber den Mechanismus des Wiederstandes, den ein bewegter Krper in einer Flussigkeit erfahrt. Nachrichte der K. Gesellschaft der Wissenschaften zu Gttingen Mathematisch-physikalische Klasse. (1911) 509-517). These 7 pages caused similar revolution like the Einsteins relativity theory. Prandl realised that the dimensional analysis can be misleading and that the last term on the right hand side of the previous equation cannot be neglected even for infinitely large Re, because viscous fluid sticks at wall and very large velocity gradients exist in a thin boundary layer. The whole flow field is to be separated to an inviscid region and to a boundary layer, described by parabolised NS equations.Prandtl L.: ber Flussigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen d.III Internat.Math.Kongress, Heidelberg 8.-13.August 1904, B.G.Teubner, Leipzig 1905, S.485-491Quite different view was suggested by Ludwig Prandtl during conference in 1904:
Boundary layerMHMT6Outside the boundary layer the velocity field is described by the Euler equation. For inviscid incompressible flow the relationship between velocities and pressure are expressed by Bernoulli theorem
Boundary layer MHMT6Inspection analysis must be carried out with a great care taking into account relative magnitude of individual terms in the Navier Stokes equations (and to distinguish magnitudes in longitudinal (1) and transversal () direction)..terms of the order can be neglected
Boundary layer MHMT6Equations describing boundary layer are reduced to The last equation shows that the pressure is constant in the transversal direction and its value is determined by Bernoulli equation applied to outer (inviscid) region, therefore the momentum equation in the x-direction is Remark: While the original Navier Stokes equations are elliptic, the boundary layer equations are parabolic, which means that we can describe an evolution of boundary layer in the direction x (this coordinate plays a similar role as the time coordinate in the time evolution problems). This is a great advantage because marching technique enables step by step solution and it is not necessary to solve the whole problem at once.
Boundary layer - separationMHMT6These equations are the basis of Prandtls boundary layer theory, which offers the following two important results:Viscous (friction) drag forces, called skin drag can be predicted. As soon as there is no separation of boundary layer from the surface the outer flow is not affected by the presence of boundary layer. The outer flow can be therefore solved in advance separately giving (via Bernoullis equation) pressure and boundary conditions for the boundary layer region.Form drag. Point of separation of the boundary layer on highly curved surfaces (cylinders, spheres, airfoils at high attack angles) can be predicted too. The separation occurs at the point with adverse pressure gradient, and the separated boundary layer forms a Helmholtz discontinuity surface. Behind the discontinuity is formed a wake (dead fluid region) increasing the form drag (drag caused by pressure imbalances) significantly.
Boundary layer - Plate MHMT6Probably the most important question is the thickness of boundary layer (x). Only qualitative answer follows from the boundary layer equation (parallel flow along a plate with zero pressure gradient)approximated very roughly asgivingThis preliminary conclusion is qualitatively correct: boundary layer thickness increases with square root of distance, kinematic viscosity and decreases with increasing free stream velocity. See also the theory of PENETRATION DEPTH.
Boundary layer - PlateMHMT6Little bit more precise analysis is based upon LINEAR velocity profile across the boundary layer. Integral balances of a rectangular control volume (height H)Continuity equationIntegral balance in the x-direction
Boundary layer - PlateMHMT6Linear velocity profile results in not very precise solutionCubic velocity profile is much better givingExact formulation of differential equation of boundary layer was presented by Blasius (Prandtls student), however this ordinary differential equation requires numerical solution.Anyway, knowing approximations of velocity profiles, it is possible to calculate viscous stresses upon the plate and therefore the drag force. For linear velocity profilelinear profilemore acccurate Blasius solution
Von Karman Integral theorem MHMT6HopperPrandtl equations of boundary layer are partial differential equations. Karman theorem derives from these equations ordinary differential equation, suitable for approximate solutions.Ordinary differential equation is obtained by integration of continuity equation across the boundary layer (in this way transversal velocity is expressed in terms of longitudinal velocity) and by integration of momentum equation in the transversal direction across the whole boundary layer.
Integral equations MHMT6Continuity equation integrated across the boundary layerMomentum equationElimination of transversal component using continuity equation
Integral equations MHMT6Integrated momentum equation can be rearranged toMomentum thickness **Displacement thickness *Displacement of surface corresponding to the same flowrate of ideal fluidIn a similar way the energy integral equation can be derived (multiply momentum equation by ux and integrate)Energy thickness ***Dissipation integral
Integral equations MHMT6Integral momentum equation holds for laminar as well as turbulent flowThe only one differential equation is not enough for solution of 3 variables: displacement thickness, momentum thickness and shear stress. Approximate solutions are based upon assumption of similarity of velocity profiles in the boundary layer.
Integral equations MHMT6The simplest example is laminar boundary layer at parallel flow along a plate. In this case U=constant and integral momentum balance reduces toBetter approximation than the previously analyzed linear velocity profile ux=Uy/ is a cubic velocity profile, because the cubic polynomial with 4 coefficients can respect 3 necessary boundary conditions corresponding to laminar flow: zero velocity at surface y=0, prescribed velocity U and zero stress dux/dy at y=.
Integral equations MHMT6Substituting this velocity profile to definition of ** and w=dux/dyKarman integral balance reduces to the ordinary differential equation for thickness of boundary layerSolution is the previously presented (but not derived) result Please mention the fact that the momentum thickness ** is much less than the boundary layer thickness
Integral equations MHMT6Turbulent boundary layer is described by the same integral equation, but it is not possible to use the same velocity profile (this is not true that duz/dy is zero at y=) and first of all the turbulent wall shear stress cannot be expressed in the same way like in laminar layer. Brutal simplification based upon linear velocity profile, and simplified Prandtls model of turbulence (=2U2) gives linearly increasing boundary layer thicknessIn reality increases more slowly, with the exponent 0.8 ( ) this prediction is based upon Blasius formula for friction factor (see textbook Sestak et al: Penos hybnosti a tepla (1988), p.94). More accurate result (see Schlichting, Gersten: Boundary layer theory, Springer, 8th edition 2000) isG only weakly depends on Re and limiting value is 1