Learning sources John D. Anderson, Jr., Fundamentals of Aerodynamics Pijush K. Kundu, Ira M. Cohen, Fluid Mechanics. 4th edition, 2008, Academic Press; Books on Fluid dynamics / Fluid mechanics / Hydrodynamics / Aerodynamics; Research papers, especially on methods of computation; http://mathworld.wolfram.com/ http://scienceworld.wolfram.com/ http://wikipedia.com/ Warning! Be careful when using any info from Wikipedia as it can be inaccurate Flow visualisation: For example http://serve.me.nus.edu.sg/limtt/#Flow_Gallery http://alexl.wordpress.com/fascinating-flows/ ES174, car Aerodynamics
Computational Fluid Dynamics (CFD) CFD software solves equations Input: Equations + Boundary Conditions + Grid parameters Output: Flow structure, Velocity Profiles, Pressure distribution Flow visualization: Streamline: A line that is always the same direction as the local flow velocity; Streak line (Filament line): The line taken up by successive particles of fluid passing through some given point; Pass line: The path traced by any one particle of the fluid in motion.
Notations Fluid Velocity, m/s t Time, s x, y, z Coordinates, m Continuity equation When fluid is treated as incompressible (Mach number << 1, subsonic flow), so that
Equations of motion: 2nd Newton’s law compare with Euler Equations (components separately) Navier-Stokes Equations: Euler Equations + viscosity
CFD approaches Full Navier-Stokes equations: computationally intensive, often unnecessarily Reynolds-Averaged Navier-Stokes (RANS): Mean Flow + Turbulent pulsations Large Eddied Simulation (LSE): a relatively new method for turbulent flows Boundary Conditions: fluid velocity, pressure Can be more complicated: waves, droplets, compliant walls etc Grid: must be of an appropriate spacing Non-uniform grids are very common
Warning! CFD is not a magic From the Album of Fluid Motion by Milton Van Dyke
Pressure: Bernoulli Law Higher velocity Lower pressure Air Density approximately 1.2 kg/m3 Back-of-the envelope estimate for the Pressure Drag force:
L h Then.. Laminar flow, turbulent flow; Reynolds number similarity: when Re is high enough, the flow structure does not change Big whorls have little whorls That feed on their velocity, And little whorls have lesser whorls And so on to viscosity. Lewis F. Richardson Lift, form drag, skin friction ..heat conduction etc Cavity flows + unsteady Boundary layers: laminar, turbulent + thickness estimates
Then.. Laminar flow, turbulent flow; Reynolds number similarity: when Re is high enough, the flow structure does not change Lift, form drag, skin friction Cavity flows + unsteady Boundary layers: laminar, turbulent + thickness estimates Turbulent boundary layer + CFD movies http://flow.kaist.ac.kr/bbs/board.php?bo_table=databaseinturbulen&wr_id=5 efluids image gallery: turbulence http://www.efluids.com/efluids/gallery/gallery_pages/1turbulence_page.jsp VOLVO car (next page) http://knol.google.com/k/-/-/yvfu3xg7d7wt/8nue4f/volvocar.jpg
Cars Does the flow separation occur in the simulation as it does in reality? http://www.fenics.org/w/upload/1/14/Volvo.jpg
Meshes Meshes http://www.scorec.rpi.edu/~garimell/blmesh.html
Viscous scale -1 S, m3/s2 Dynamic range S = 2/3K-5/3 Energy scale L-1 k, m -1 Then.. Full Navier-Stokes equations: computationally intensive, often unnecessarily intermediate Large Eddied Simulation (LSE): a relatively new method for turbulent flows Reynolds-Averaged Navier-Stokes (RANS): Mean Flow + Turbulent pulsations Issues: mesh, numerical precision, time dependence (unsteady cavity flow) Unsteady cavity flow http://www.hector.ac.uk/casestudies/circular_cavity_flows.php
j-2 j-1 j j+1 i+1 i i-1 x t (time) Numerics: discretisation Vj Vj+1 Euler method: 1st order, i.e. error proportional to 2 Runge-Kutta 45 method: better than 4th order, i.e. error proportional less 4
inlet: normal velocity outlet: pressure walls: smooth, zero normal velocity (impermeable) Numerics: boundary conditions Turbulence intensity is assumed 2% everywhere (may be varied). This defines the Turbulent Viscosity (Eddy Viscosity). Step-by-step instructions (will be posted to the course website): http://go.warwick.ac.uk/fluidseminar/es174_cfd_2008.pdf
f(t) t (time) f(t) t (time) Numerics: time dependence Flow simulation toolbox: N-S equations are solved as time-dependent until the flow stabilises. When a chosen parameter reaches a constant value, computation terminates. In reality, this may never happen by either physical or numerical reasons. When numerical instability occurs, the artificial viscosity is introduced in the form of higher order derivatives: 4V/ x 4 etc.
More pictures: unsteady flow Helmholtz instability http://www.iag.uni-stuttgart.de/people/ andreas.babucke/work.html Instability of a smoke jet Instability of a jet: http://alexl.wordpress.com/fascinating-flows/
inlet: normal velocity outlet: pressure walls: smooth, zero normal velocity (impermeable) Numerics: general Step-by-step instructions (will be posted to the course website): http://go.warwick.ac.uk/fluidseminar/es174_cfd_2008.pdf A general advise: Do not pay too much attention to detailed sketching of the car. Instead, try to understand results of the simulation for a relatively simple shape: ● plot streamlines, try to understand if the flow separation occurs; ● plot velocity and pressure profiles along and across the tunnel; ● calculate the pressure force acting on upstream-facing panels … etc.