Chapter 1 Introduction of boundary layer phenomena T-S Leu Jan. 13, 2018 Main Topics • History of Fluid Mechanics Development • Idea of Boundary Layer • Boundary Layer Equations 1 Fluid Mechanics Development History EFD: Ideal fluid: Real fluid : Hot wire MEMS Inviscid flow (zero drag) Hot film Viscosity PIV 2000s Lagrangian/Eulerian Viscous flow Boundary LDV MD Potential flow Laminar/ layer 1990s 1980s DNS (Inviscid+Irrotational) Turbulent flow Ludwig 1960~70s CFD Osborne Prandtl Panel Turbulence Reynolds 1904 method modeling 1842~1912 Lagrange Richard (1827~1845) (1736~1813) 1842~1919 Feynman has Navier Laplace Lord described Cauchy Jean (1749~1827) Rayleigh turbulence as Poisson D'Alembert Hydrodynamic the most St. Venant Isaac 1752 1755 instability: important Stokes Newton 1738 Leonhard Taylor Rayleigh unsolved 1687 Daniel Euler Navier- Kelvin Helmholtz problem of Bernoulli Stokes Benard cells classical physics. 89y Equation Bernoulli Equation 2 Bernoulli equation (I) BERNOULLI’S EQUATION (II) & integration BERNOULLI’S EQUATION AERODYNAMICS (W2-1-2) 3 BERNOULLI’S EQUATION FOR AN IRROTATION FLOW incompressibl e Steady Irrotational No gravity AERODYNAMICS (W2-1-2.1) Unsteady Bernoulli Equation This is not a very useful result in general since ∂vs/∂t can change dramatically from one point to another; to use this in practice we need to be able to draw streamline shapes at each instant in time. It works especially for simple cases such as impulsively started confined flows where streamlines have the same shape at each instant and we are interested in time required to start the flow. 4 Exercise 1 Flow out of a long pipe connected to a large reservoir, (1)find the steady state velocity v2 in the pipe after the the transient stage (2)find the transient velocity v2 in the pipe changing with time during the transient stage Back to History D'Alembert's paradox •Influid dynamics, d'Alembert's paradox is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert. • D'Alembert proved that for incompressible and inviscid potential flow –thedrag force is zero on a body moving with constant velocity relative to the fluid. • Zero drag is in direct contradiction to the observation of substantial drag on bodies moving relative to fluids, such as air and Jean le Rond d'Alembert (1717- water; especially at high velocities 1783) corresponding with high Reynolds numbers. https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox Back to History 5 Lagrangian/Eulerian Description in Fluid Mechanics Assignment: Watch the video about Lagrangian/Eulerian Description in Fluid Mechanics https://www.youtube.com/watch?v=mdN8OOkx2ko (MP4) METHODS OF DESCRIPTION • Lagrangian description => System • Eulerian description => Control volume Ch 1-2 6 Lagrangian Description • Attention is focused on a material volume (MV) and follow individual fluid particle as it move. • The fluid particle is colored, tagged or identified. • Determining how the fluid properties associated with the particle change as a function of time. Example: one attaches the temperature-measuring device to a particular fluid particle A and record that particle’s temperature as it moves about. TA = TA(t)=T (xo,yo,zo, t) where particle A passed through coordinate (xo,yo,zo) at to The use of may such measuring devices moving with various fluid particles would provide the temperature of these fluid particles as a function of time. Ch 1-2 Eulerian Description • Attention is focused on the fluid passing through a control volume (CV) fixed in the space. • Obtaining information about the flow in terms of what happens at the fixed points in space as the fluid flows past those points. • The fluid motion is given by completely prescribing the necessary properties as a functions of space and time. Example: one attaches the temperature-measuring device to a particular point (x,y,z) and record the temperature at that point as a function of time. T = T ( x , y , z , t ) => field concept. The independent variables are the spatial coordinates ( x , y , z) and time t Ch 1-2 7 Field Representation of flow • At a given instant in time, any fluid property ( such as density, pressure, velocity, and acceleration) can be described as a functions of the fluid’s location. This representation of fluid parameters as functions of the spatial coordinates is termed a field representation of flow. • The specific field representation may be different at different times, so that to describe a fluid flow we must determine the various parameter not only as functions of the spatial coordinates but also as a function of time. • EXAMPLE: Temperature field T = T ( x , y , z , t ) • EXAMPLE: Velocity field V u(x, y, z,t) i v(x, y, z,t) j w(x, y, z,t) k Ch 1-2 Nature and Transformation of Lagrangian and Eulerian Description • It is more nature to apply conservation laws by using Lagrangian description (ie. Material Volume). • However, the Eulerian description (ie. Control Volume) is preferred for solving most of problem in fluid mechanics. • The two descriptions are related and there is a transformation formula called material, total or substantial derivative between Lagrangian and Eulerian descriptions. Ch 1-2 8 Material Derivative (I) • Let (x,y,z,t) be any field variable, e.g., ρ, T, V=(u,v,w), etc. (Eulerian description) • Observe a fluid particle for a time period t as it flows (Langrangian description) • During the time period, the position of the fluid particle will change by amounts x , y , z, while its vale of will change by an amount t x y z t x y z x y z • As one follow the fluid particle, ( , , ) (u,v, w) So t t t D x y z lim u v w Dt t0 t t x t y t z t t x y z which is called the material, total, or substantial derivative. Ch 1-3 Material Derivative (II) • Use the notation D/Dt to emphasize that the material derivative is the rate of change seen by an observer “following the fluid.” • The material derivateive express a Langrangian derivative in terms of Eulerian derivatives. • In vector form, D u v w (V ) Dt t x y z t • May also use the “index notation” and Einstein’s “summation convention” (i.e, summing over repeated indices) to write D uk Dt t xk Where (x1,x2,x3)≡(x,y,z) and (u1,u2,u3)≡(u,v,w) Note: The repeated index that us summed over is called a “dummy index”; one that is not summed is called a “free index”. Back to History 9 Potential Flow Theory Inviscid & Irrotational flow GOVERNING EQUATION FOR IRROTATIONAL, INCOMPRESSIBLE FLOW: LAPLACE’S EQUATION Continuity equation Incompressible: constant For incompressible flow: there exists a streamfunction For irrotational flow: there exists a velocity potential Laplace’s equation y= 0 For irrotational, incompressible flow, there are velocity potential and For irrotational, incompressible streamfunction that both satisfy flow: y Laplace’s equation. 2 2 2 0 x2 y2 2 2 2 0 Laplace’s equation x2 y2 AERODYNAMICS (W2_1_6) 10 Potential Flow Theory • Governing equation for Potential flow is Laplace equation 2 2 2 0 x2 y2 • Laplace’s equation is a second-order linear partial differential equation. If 1, 2, 3, … , n represent n separate solutions of Laplace’s equation, then =1+2+3+… +n is also a solution of Laplace’s equation. • Complex potential with conformal mapping Boundary Condition for LAPLACE’S EQUATION Boundary Conditions: Infinite boundary conditions: Wall boundary conditions: or or 11 Fluid Flow Governing Equations • Mass conservation => Continuity equation • Momentum equation F=ma=d(mV) /dt => Navier Stokes Equation •1st Thermaldynamic law (Conservation of Energy) => Energy equation Fluid Flow Governing Equations • Continuity equation: D uk (uk ) 0 t xk Dt xk • Momentum (N-S) equation: u u p u u u j j k i j [ uk ] ( ) f j t x x x x x x x k j j k i j i • Energy equation: e e uk T ( uk ) p (k ) t xk xk xi x j u u u u ( k )2 i j j xk x j xi xi 12 2 Navier-Stokes Equations Full N-S equation 3 u u p u u u j j k i j [ uk ] ( ) f j t x x x x x x x k j j k i j i u For incompressible flows, V k 0 xk u u p u u j j i j [ uk ] f j t x x x x x k j i j i For incompressible flows with constant viscosity , u u p u 2u p 2u j j i j j [ uk ] f j f j t x x x x 2 x 2 k j j i xi j xi For incompressible ,inviscid fluids =0, uj uj p [ uk ] f j The Euler equations t xk x j e e uk T ( uk ) p (k ) t xk xk x j x j u u u u ( k )2 i j j e C T v xk x j xi xi T T u 2T k for constant C & k Cv ( uk ) p k 2 v t xk xk x j 2 T T T for incompressible flow Cv ( uk ) k 2 with constant Cv & k t xk x j 2 T T uk T for inviscid flow with Cv ( uk ) p k 2 constant Cv & k t xk xk x j T T 2T for incompressible inviscid Cv ( uk ) k 2 flow with constant C & k t xk x j v 13 where where 14 Molecular and Statistical Approaches • Fluids consist of molecules whose motion is governing by the law of dynamics. • The macroscopic phenomena are assume to arise from the molecular motion of the molecules.