Hydrodynamic Instability. Agenda. Problem definition. Heuristic approach: Intuitive solution. Mathematical approach: Linear stability theory

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1 Environmental Fluid Mechanics II Stratified Flow and Buoyant Mixing Hydrodynamic Instability Dong-Guan Seol INSTITUTE FOR HYDROMECHANICS National Research Center of the Helmholtz Association Agenda Problem definition Heuristic approach: Intuitive solution Mathematical approach: Linear stability theory

2 Problem definition: Kelvin-Helmholz instability Kelvin-Helmholz instability: product of competion between density stratification and velocity shear! Problem definition: Kelvin-Helmholz instability Kelvin-Helmholz instability in shear flow

3 Vortical structures in the Mekong River near Phnom Penh, Cambodia Simulation of Kelvin-Helmholz instability

4 Objective and Causes of instability Objective: When and How stable flows break down? Causes of instability Unstable velocity profiles Unstable stratification (heavy fluid above the light fluid) Combination of velocity shear and stratification Approaching methods to the problem Heuristic approach: Intuitive solution Mathematical approach: linear stability analysis Heuristic approach Intuition: Instability is expected when the available kinetic energy of the shear flow exceeds the buoyant work stabilizing destabilizing W B2 Density Velocity y+ y W B

Buoyant work Consider the buoyancy change of the heavier fluid parcel To get the buoyant work, let us integrate the buoyant force along the path Then, the total buoyant work is given by 9 10.05.

5 Buoyant work Consider the buoyancy change of the heavier fluid parcel To get the buoyant work, let us integrate the buoyant force along the path Then, the total buoyant work is given by Kinetic energy Initial kinetic energy Boussinesq approximation Final kinetic energy Mean velocity - Upper layer decelerate - Lower layer accelerate Kinetic energy change Finally, instability takes place when kinetic energy exceeds buoyancy work

6 Mathematical approach: Linear Stability Analysis Basic Assumption o Fluids are immiscible o Fluids are inviscid o Flow in each layer is irrotational o Applicable linearized theory Governing equations: Inviscid Navier-Stokes equations Continuity equation Momentum equation Solution procedure 1. Start with the linear solution for a flow field 2. Derive the equations for small disturbances (linear waves) 3. Substitute disturbance equations into the governing equations and boundary conditions of fluid motion to obtain an eigenvalue problem 4. Solve the eigenvalue problem

7 Step 1. Linear Solution for each layer o Layer 1. o Layer 2. Where : pressure at z=0 (interface) Step 2. Add disturbance to the linear solution disturbances Step 3.1 Substitute disturbances to the governing equations o Continuity equation - Layer 1 - Layer 2 o Momentum equations (recall ) - Layer 1 - Layer

8 Step 3.2 Substitute disturbances to the boundary conditions o Far-field boundary conditions - Layer 1 - Layer 2 o Kinematic boundary conditions : Fluid parcels move tangentially to the interface (immiscible) Function of interface - Layer 1 - Layer 2 o Dynamic boundary condition: Pressure is continuous at the interface Now, we have 2 governing equations and 3 boundary equations for each layer Continuity Layer 1 Layer 2 Momentum BC KBC DBC These equations are linear so that the solutions can be in the form of trigonometric functions

9 Step 3.3 Introduce trigonometric functions o Disturbances Linearization o Then, the unknowns can be expressed as Plug these into the governing equations and BCs to obtain the eigenvalue problem Layer 1 o Momentum eqn. o KBC o BBC Layer 2 o Momentum eqn. Finally, we have5 equations and 5 unknowns o KBC

10 Step 4. Solve Eigenvalue problem To have non-trivial solution, the coefficient matrix should have zero for its determinant Solve for Criteria for stability: substitute the solution to the disturbance wave Damps: stable Neutral Growth: unstable

11 Criteria for stability Then, the criteria for the stability becomes Introduce characteristic length scale L as vertical depth and using Taylor s expansion and Boussinesq approximation where : reference density Criteria for stability Substitute to the criteria for the stability : Stable : Unstable : Billows start break

12 Critical wave length From the criteria for the stability, Plug the following into the above, ΔU = U 0 = U 1 U 2 ρ 1 ρ 2 = Δρ ρ 1 + ρ 2 = ρ 0 + ρ 0 =2ρ r k 2gΔρρ r ΔUρ 2 r L πδuρ r gδρ L< πδuρ r gδρ : Stable : Unstable

Prototype Instabilities

Prototype Instabilities David Randall Introduction Broadly speaking, a growing atmospheric disturbance can draw its kinetic energy from two possible sources: the kinetic and available potential energies