14. Computational Fluid Dynamics CH EN 374: Fluid Mechanics

Where We’ve Been 1rst Exam Period Microscopic (differential) balances: Viscosity, shear stress Hydrostatic pressure, pressure forces Macroscopic (integral) materials balances: Mass Momentum (and force) Energy Microscopic (differential) balances: Mass – continuity equation Momentum (and force) – Navier-Stokes NS Approximations and types of flow Creeping (Stokes) flow Inviscid flow Boundary layers

Today Computational Fluid Dynamics (CFD) Most CFD is performed using specific software packages that take time to learn So we’re not going to do it ourselves in this class Today I will introduce you to important CFD concepts

Computational Fluid Dynamics Use numerical methods to solve the equations of motion: Continuity X-momentum Y-momentum Z-momentum Four equations, 4 unknowns ( 𝑣 𝑥 , 𝑣 𝑦 , 𝑣 𝑧 , 𝑃) What if heat transfer is important in our flow? We might want to predict temperature, 𝑇 Any ideas about where we’d get another equation?

Advantages of CFD Example: Biomedical Engineering/Medicine Discuss: https://www.youtube.com/watch?v=Vw_lGHrs0Fk https://www.youtube.com/watch?v=Y-gqQXegJvY Discuss: What are some of the advantages of using CFD over only using experimental results? Why is it important to also have experimental data? How does this apply to areas other than medicine?

Algorithm Example So ho does CFD solve problems we can’t solve analytically? Divide flow into a grid of points 𝑖 𝑖+1 𝑖−1 𝑖− 1 2 𝑖+ 1 2

Algorithm Example 𝑖 𝑖+1 𝑖−1 𝑖− 1 2 𝑖+ 1 2 𝜕 2 𝑣 𝑥 𝜕 𝑥 2 𝑖 ≈ 𝜕 𝑣 𝑥 𝜕𝑥 𝑖+1/2 _ 𝜕 𝑣 𝑥 𝜕𝑥 𝑖−1/2 Δ𝑥 𝜕 𝑣 𝑥 𝜕𝑥 𝑖+1/2 ≈ 𝑣 𝑥 𝑖+1 − 𝑣 𝑥 𝑖 Δ𝑥 𝜕 𝑣 𝑥 𝜕𝑥 𝑖−1/2 ≈ 𝑣 𝑥 𝑖 − 𝑣 𝑥 𝑖−1 Δ𝑥 𝜕 2 𝑣 𝑥 𝜕 𝑥 2 𝑖 ≈ 𝑣 𝑥 𝑖+1 −2 𝑣 𝑥 𝑖 + 𝑣 𝑥 𝑖−1 Δ 𝑥 2

Algorithm Example 𝑖 𝑖+1 𝑖−1 𝑖− 1 2 𝑖+ 1 2 We have seen how to re-write NS terms as approximations of points on a grid Based on what you know about numerical methods (263!) how would you describe what needs to happen next?

Grids Flow domain is divided into cells (a grid) Boundary conditions must be the same when boundaries touch Equation solved for each cell

Gridding/Meshing What do you think the advantages and disadvantages of a finer grid (smaller cells) would be?

Boundary Conditions Boundary conditions defined on outer edge of computational domain. Types of boundary condition: Wall Inflow/outflow Symmetry Internal

Turbulent Flow Direct Numerical Simulation (DNS) Directly calculates every tiny eddy from NS What kind of grid do you think this requires? What do you think the advantages/disadvantages of this would be? https://www.youtube.com/watch?v=hz7UjN_vYuw&t=52s

Turbulent Flow Large Eddy Simulation (LES) NS for big eddies Simpler model (an additional equation of motion) for smaller eddies What do you think the advantages/disadvantages of this would be? https://www.youtube.com/watch?v=iELFWDuSoe4

Results: Visualization Methods Streamlines

Results: Visualization Methods Contour Plots

Results: Visualization Methods Vector Plots

Example Questions About CFD Given this picture describes the results of a CFD simulation, name the visualization method. Name some tradeoffs to consider when meshing a system for CFD. Why are turbulent flows more difficult to simulate?

Examples! https://www.youtube.com/watch?v=hEsmlS6EGJw https://www.youtube.com/watch?v=sdYmJ2SJcZE https://www.youtube.com/watch?v=-9FwoFjDt7s https://www.youtube.com/watch?v=D6iuVr9V6os https://www.youtube.com/watch?v=7x3Fb9HwmRE