lecture6 - 2.20 - Marine Hydrodynamics, Spring 2005 Lecture...

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± Lecture 6 - Marine Hydrodynamics Lecture 6 2.2 Similarity Parameters from Governing Equations and Boundary Conditions In this paragraph we will see how we can specify the SP’s for a problem that is governed by the Navier-Stokes equations. The SP’s are obtained by scaling , non-dimensionalizing and normalizing the governing equations and boundary conditions. 1. Scaling First step is to identify the characteristic scales of the problem. For example: Assume a flow where the velocity magnitude at any point in space or time | ± x, t )i s about equal to a velocity U , i.e. | v ( ± | v ( ± | ± x, t )= αU , where α is such that 0 α O (1). Then U can be chosen to be the characteristic velocity of the flow and any velocity ±v can be written as: = U±v ± where it is evident that ± is: (a) dimensionless (no units), and (b) normalized ( | ± |∼ O (1)). Similarly we can specify characteristic length, time, pressure etc scales: Characteristic scale Dimensionless and Dimensional quantity normalized quantity in terms of characteristic scale Velocity U ± = ± Length L ±x ± = L±x ± t ± Time T t = Tt ± Pressure p o p v p p =( p o p v ) p ± 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20
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2. Non-dimensionalizing and normalizing the governing equations and bound- ary conditions Substitute the dimensional quantities with their non-dimensional expressions (eg. substitute ±v with U±v ± , ±x with L±x ± , etc) into the governing equations, and boundary conditions. The linearly independent, non-dimensional ratios between the character- istic quantities (eg. U , L , T , p o p v ) are the SP’s. (a) Substitute into the Continuity equation (incompressible flow) ∇· =0 U ± ± L ± · ± Where all the () ± quantities are dimensionless and normalized (i.e., O(1)), v for example, ∂² ± = O (1). ∂x ± (b) Substitute into the Navier-Stokes (momentum) equations ∂±v 1 +( ·∇ ) = − ∇ p + ν 2 g ˆ j ∂t ρ U∂±v ± U 2 p o p v νU + ( ± ± ) ± = ± p ± + ( ± ) 2 ± g ˆ j T ± L ρU 2 L 2 divide through by U L 2 , i.e., order of magnitude of the convective inertia term L ± ∂± ² ± ± ² ± p ν ³ ± g L v o p v ± ´ µ ) = ( p )+ 2 ˆ j UT ρU 2 UL U 2 The coefficients ( ) are SP’s. 2
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± Since all the dimensionless and normalized terms () ± are of O(1), the SP’s ( ) measure the relative importance of each term compared to the con- vective inertia. Namely, L Eulerian inertia ∂²v S = Strouhal number ∂t UT convective inertia ( ±v ·∇ ) The Strouhal number S is a measure of transient behavior. For example assume a ship of length L that has been travelling with velocity U for time T . If the T is much larger than the time required to travel a ship length, then we can assume that the ship has reached a steady-state.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.

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lecture6 - 2.20 - Marine Hydrodynamics, Spring 2005 Lecture...

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