Fluid Flow: Unsteady Flow
Objectives Section 5 – Fluid Flow Module 5: Unsteady Flow Page 2 • Understand unsteady flow. • Examine the unsteady form of the Navier–Stokes Equation. • Study the Courant Number for unsteady flow. • Learn from an example: Unsteady flow past a cylinder
Unsteady Flow: Part I Section 5 – Fluid Flow Module 5: Unsteady Flow Page 3 • Flow conditions such as pressure, velocity or even domain boundaries change with time. • Relatively complex and time consuming to solve. • It is important to choose a reasonable “time step” size. • The rate of progress in time depends upon the steepness of the time gradient. • The smaller the time step, the higher the stability and accuracy. • However, more computational resources are required with smaller steps. Accuracy Stability Computational Resources
Unsteady Flow: Part II Section 5 – Fluid Flow Module 5: Unsteady Flow Page 4 • Time steps can be as large as hours, days or months when studying effect of anthropogenic greenhouse gases in Earth’s atmosphere. • Time steps can be microseconds when studying shock wave phenomena. • In order to work out an ideal time step, the Courant Number is generally kept under one (for implicit scheme). • All turbulent flows are essentially unsteady, laminar flows on the other hand can be steady as well as unsteady. • A good example of unsteady flow is seen in reciprocating devices such as piston engines and compressors.
Unsteady Flow: Part III Section 5 – Fluid Flow Module 5: Unsteady Flow Page 5 • Consider the example of a four stroke internal combustion engine, focusing on the exhaust cycle. • As the piston rises and the valve opens, the flow velocity at the outlet valve increases, approaches a maximum and then decreases. • This is one of the most common cases of unsteady flow that is solved by CFD. • It gives an insight into flow that is difficult to get through prototype testing. • This information is used to design the exhaust manifold. • Setting up the right time-step is crucial.
Navier–Stokes Equation Section 5 – Fluid Flow Module 5: Unsteady Flow Page 6 • Time derivative takes prominence and the strength of the time derivative will dictate the size of the time step used for analysis. • The time step refers to the size of the leap in temporal domain when progressing in time. • The higher the time derivative, the smaller the time step for numerical analysis.
Courant Number Section 5 – Fluid Flow Module 5: Unsteady Flow Page 7 • Also termed as the Courant–Friedrichs–Lewy condition (CFL condition), the Courant Number is a useful tool to evaluate the time step size for unsteady flow cases. • The Courant Number for a 1 dimensional case can be given as: • The Courant Number for a 2 dimensional case can be given as: • Where: • u is the flow speed, • Δt is the time step • Δx is the grid size For a solution to be stable, the Courant number should be less than or equal to 1
Example:Flow Across a Cylinder: Part I Section 5 – Fluid Flow Module 5: Unsteady Flow Page 8 • Flow shows different behavioratdifferent flow speeds. • A Von Karman Vortex Street (a fascinating phenomenon for fluid flow enthusiasts) can be observed at 40 < Re < 200,000. • Re = Reynolds Number • A two-part video for this module on unsteady flow covers setting up, solving and viewing results for a Von Karman Vortex Street. y x
Example:Flow Across a Cylinder: Part II Section 5 – Fluid Flow Module 5: Unsteady Flow Page 9 • Various patterns of flow across a cylinder are seen at different Reynolds numbers. • Creeping Flow Re<10 • Attached Vortices 10<Re<40 • Von Karman vortex trail 40<Re<200,000 • Fully turbulent wake Re>200,000
Summary Section 5 – Fluid Flow Module 5: Unsteady Flow Page 10 • Unsteady fluid flows occur in a variety of real life situations. • From starting and stopping phases of engines and wind turbines to vehicles for transport, unsteady flow patterns are frequently encountered. • When analyzing flow variation with time, the analysis is conducted for a series of time steps, which progress in time. • The size of these time steps depends upon the time gradient. • The higher the time gradient, the lower the time step should be.
Summary Section 5 – Fluid Flow Module 5: Unsteady Flow Page 11 • The Courant Number helps estimate the proper time step. • If the duration of the time step is very low, the amount of computation required becomes extremely high. • Thus a compromise must be sought which depends upon the available time and computer resources as well as required accuracy.