Introduction to Fluid Dynamics & Applications Nitin Jain COMP 259 Class Presentation The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL.
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Introduction to Fluid Dynamics & Applications Nitin Jain COMP 259 Class Presentation The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Motivation • Starting point for all simulation and animation that involves fluid flow : indispensable for physically based modeling 2 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Overview • Understanding & Deriving the NavierStokes equations • Solving the Navier-Stokes equations - Basic approach - Boundary conditions • Applications - Lid Driven Cavity flow - Smoke 3 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL So, Let’s begin the journey… 4 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Foundations • Physical Quantities - 5 u P d g : : : : The velocity field Pressure Density Gravity & other external forces The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Foundations Operators - gradient div 2 6 u ,∂ u u ∂ x ∂ y ∂ - divergence div u u u - Laplacian 2u u 2 x x u 2 u 2 y y 2 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Navier-Stokes Equations u 0 1 du 2 (u )u p u f dt convection 7 pressure viscosity external forces The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Navier-Stokes Equations • In two dimensions … 8 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Boundary Conditions • • • • • 9 No-slip condition Free-slip condition Inflow condition Outflow condition Periodic Boundary Condition The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Derivation • Transport Theorem • Conservation of Mass • Conservation of Momentum 10 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Transport Theorem (c , t ) c 0 t d f ( x , t )dx f div( fu )( x, t )dx t t dt t x (c , t ) 11 u ( x , t ) (c , t ) t The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Conservation of Mass mass ( x,0)dx ( x, t )dx 0 t ; is density Transport theorem d ( x, t )dx div(u )( x, t )dx 0 t t dt t div( u ) 0 t is constant for incompressible fluids Integrand vanishes div u 0 Continuity equation 12 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Conservation of Momentum momentum t ( x, t )u ( x, t )dx changein momentum actingforces body forces: ( x, t ) f ( x, t )dx t surface forces: σ( x, t )nds t σ : stress tensor n : normal d dt 13 t ( x , t )u ( x , t )dx ( x , t ) f ( x , t )dx σ( x , t )nds t t The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Conservation of Momentum d dt t ( x , t )u ( x , t )dx ( x , t ) f ( x , t )dx σ( x , t )nds t t Transport theorem Divergence theorem d ( u ) (u )( u ) ( u ) div u g div σ 0 dt … 1 du 2 (u )u p u f dt Momentum equation 14 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Dynamic Similarity of flows • Dimensionless quantity = dimensional quantity/ reference quantity with same physical units • The reference quantities used should possess certain properties : - They should be constant for the problem - They should be known in advance - They should be characteristic for the problem 15 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Dimensionless Variables In a wind tunnel, L = length of the obstacle in the flow, u’,P’and p’ are the upstream velocity, pressure and density. • • • • 16 x’’ = x/L t’’ = u’t/L u’’ = u/u’ P’’ = (P-P’) / (p’u’2) The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Reynolds Number and Froude number • These numbers describe the properties of the flow - Re = p’u’L/μ - Fr = u’/sqrt(L*magnitude(g)) • The Reynolds number represents the relative magnitude of the inertial and viscous forces • The Froude number relates the velocity of object to velocity of fluid waves. 17 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Let us now see how to solve the Navier Stokes equations… 18 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Discretization • Refers to passing from a continuous problem to one considered at only a finite number of points. • Reduces differential equations to a system of algebraic equations. • The solution is found only at a finite number of points in the domain. 19 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Discretization • Forward difference - Derivative at grid point i = ((function value at i+1) – (function value at i)) / (grid spacing) • Backward difference - Derivative at grid point i = ((function value at i) – (function value at i-1)) / (grid spacing) • Central difference - Derivative at grid point i = ((function value at i+1) – (function value at i-1)) / (2*grid spacing) 20 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Pros and Cons • With forward and backward differencing, the order of approximation is first order. With central differencing, it is second order. • For convection-diffusion problems, stability problems occur with central differencing when the grid spacing is too coarse. Maintaining stability hence requires restricting the grid spacing to be very small. • A possible compromise is to use a weighted average of both discretizations : - μ*forward difference + (1- μ)*backward difference 21 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Solving the equations Basic Approach 1. Create a tentative velocity field. a. Finite differences b. Semi-Lagrangian method (Stable Fluids [Stam 1999]) 2. Ensure that the velocity field is divergence free: a. Adjust pressure and update velocities b. Projection method 22 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Staggered Grid pi , j 1 vi , j 1 2 ui 1 , j pi 1, j 2 ui 1 , j 2 pi , j vi , j 1 2 pi , j 1 pi 1, j The staggered grid provides velocities immediately at cell boundaries, is convenient for finite differences, and avoids oscillations. Consider problem of a 2D fluid at rest with no external forces. The continuous solution is: u0 v0 p constant On a discretized non-staggered grid you can have: ui , j 0 vi , j 0 pi , j P1 for i j even,P2 for i j odd 23 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The algorithm • Perform time discretization of momentum equations. Discretize in a manner explicit in the velocities and implicit in the pressure. • Evaluate continuity equation at time t(n) + 1, to yield a poisson equation for pressure. Solve for the pressure. • Use the pressure values to compute new velocities. 24 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Solving for pressure ( n1) ~ ( n ) du u u 1 p ( n1) dt t Discretize in time ( n1) ~ ( n ) t u u p ( n1) Rearrange terms t u ( n1) u~ ( n ) 2 p ( n 1) 0 25 Satisfy continuity eq. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Solving for pressure We end up with the Poisson equation for pressure. p 2 ( n 1) t u~ ( n ) This is another sparse linear system. These types of equations can be solved using iterative methods. Use pressures to update final velocities. ( n1) ~ ( n ) t u u p ( n1) 26 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL No-slip condition • The velocities vanish at the boundaries to satisfy the no-slip condition. u0,j = 0, vi,o = 0, ui-max,j = 0, vi,j-max = 0 27 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Free Slip Condition • The velocity component normal to the boundary should vanish along with the derivative of the velocity component tangent to the boundary. u0,j = 0, vi,o = 0, ui-max,j = 0, vi,j-max = 0, v0,j = v1,j , vi-max + 1,j = vi-max ,j , ui,0 = ui,1 , ui,j-max+1 = ui,j-max 28 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Outflow Condition • The normal derivatives of both velocity components are set to zero at the boundary u0,j = u1,j , ui-max,j = ui-max -1 ,j , vi-max+1,j= vi-max,j, v0,j = v1,j, ui,0 = ui,1 , ui,j-max+1 = ui,j-max vi,0 = v1,j , vi,j-max = vi,j-max-1 , 29 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Inflow Conditions • The velocities are explicitly given • We impose this for velocities normal to the boundary by directly fixing the values on the boundary line • For the velocity components tangential to the boundary, we achieve this by averaging the values on either side of the boundary. 30 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Periodic boundary conditions • For periodic boundary conditions in the x-direction, the boundary values on the left and the right boundaries coincide. u0,j = ui-max-1,j , ui-max,j = u1,j v0,j = vi-max-1,j , vi-max,j = v1,j 31 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Stability condition Limits on time step • CFL conditions – don’t move more than a single cell in one time step umax t x, vmax t y • Diffusion term 1 1 2t 2 2 y x 32 1 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Satisfying the Continuity Eq. The tentative velocity field is not necessarily divergence free and thus does not satisfy the continuity equation. Three methods for satisfying the continuity equation: 1. 2. 3. 33 Explicitly satisfy the continuity equation by iteratively adjusting the pressures and velocities in each cell. Find a pressure correction term that will make the velocity field divergence free. Project the velocities onto their divergence free part. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Explicitly Enforcing u=0 Since we have not yet added the pressure term, we can use pressure to ensure that the velocities are divergence free. u>0 increased pressure and subsequent outflux u<0 decreased pressure and subsequent influx Relaxation algorithm 1. Correct the pressure in a cell 2. Update velocities 3. Repeat for all cells until each has u<ε 34 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Let’s move on to some applications now… 35 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Lid Driven Cavity Flow • We simulate a driven cavity flow in a square domain. • The physical configuration consists of a square container filled with a fluid. • The lid of the container moves at a given, constant velocity, thereby setting the fluid in motion. 36 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Boundary Conditions • No slip conditions on all four segments of the boundary, except the upper boundary. • Vertical velocity at the upper boundary is zero. • The horizontal velocity is set equal to the given lid velocity. 37 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Lid Driven Cavity Flow • Demo time : http://www.math.ubordeaux.fr/Math_Appli/Fenetre 2/videotheque/DNS/Welcome.ht ml 38 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Smoke (Fedkiw, Stam, Jensen 1999) • Uses Euler equations instead of the Navier Stokes equations since - These are more appropriate for gas models - Are less computationally expensive • Uses “vorticity confinement” to reduce numerical dissipation 39 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Euler Equations • Incompressible Euler Equations • Solve the equation in two steps - Compute intermediate velocity without the pressure term - Forve the velocity field to be incompressible 40 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Euler Equations • Solve the second equation over a timestep, ignoring the pressure term : • Compute the pressure from the following poisson equation 41 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Euler equations • The intermediate velocity is then made incompressible by subtracting the gradient of the pressure from it. • The temperature and density of the smoke are then calculated by simply advecting along the velocity field. 42 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Vorticity Confinement • Smoke usually consists of velocity field having a significant amount of rotational and turbulence components. • Numerical dissipation damps out these phenomena, and so they are added back by a technique called vorticity confinement 43 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Vorticity Confinement • Vorticity is the curl of the velocity field. • First, calculate normalized vorticity location vectors. • Then add vorticity terms at these locations. 44 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Some Videos… 45 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL