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FLUID MECHANICS III

Equations of fluid motion and related topics 1. Navier-Stokes equations 2. Similarity laws in fluid problems

http://www.homepages.ucl.ac.uk/~uceseug/Fluids3/

Navier-Stokes Equation

1821: Navier-Stokes equations I

• Famous French bridge-builder • Constructed models for fluids and solids as aggregation of interacting particles

“The theory ... cannot suit the ordinary cases ... the results of experience are our only guide.” Claude Louis Marie Henri C.Navier, 1822 Navier (1785–1836)

1. BOUNDARY LAYER 1 Navier-Stokes Equation

1845: Navier-Stokes equations IV

• Professor of Mathematics at Cam- bridge • ”Re-discovered” equations of mo- tion of viscous fluid from con- sideration of internal fluid fric- tion adding deeper physical in- sight, presented more general form of equations George Gabriel Stokes (1819-1903)

1. BOUNDARY LAYER 2

. Navier-Stokes Equations

2D, incompressible, rectangular Cartesian coordinates:

∂u ∂u ∂u 1 ∂P ∂2u ∂2u + u + v = − + X + ν + ∂t ∂x ∂y ρ ∂x  ∂x2 ∂y2 

∂v ∂v ∂v 1 ∂P ∂2v ∂2v + u + v = − + Y + ν + ∂t ∂x ∂y ρ ∂y  ∂x2 ∂y2  ∂u ∂v + =0 ∂x ∂y

(x,y)– coordinates; (u, v)– velocity components; P – pressure; (X,Y )– components of volume forcea; ρ– density; ν– kinematic viscosity

ain the gravity field if y is in the vertical up direction X = 0 and Y = −g

1. BOUNDARY LAYER 3 Nondimensional Equations

• “Small” and “large” are meaningful only for non-dimensional values

U
























































































































































































L

Parametersa: L– length; U– velocity; ρ– density; ν– kinematic viscosity Nondimensional variables:

∗ ∗ ∗ ∗ 2 ∗ L ∗ (x, y)= L (x , y ); (u, v)= U (u ,v ); P = ρU P ; t = t U Change of variables: ∗ ∗ ∗ ∂u U 2 ∂u ∂u U ∂u ∂2u U ∂2u = ; = ; = ; ∂t L ∂t∗ ∂x L ∂x∗ ∂x2 L ∂x∗2 ···

aGravity is neglected

1. BOUNDARY LAYER 4

Nondimensional Equations

Navier-Stokes equations become:

∗ ∗ ∗ ∗ 2 ∗ 2 ∗ ∂u ∗ ∂u ∗ ∂u ∂P 1 ∂ u ∂ u + u + v = − + + ; ∂t∗ ∂x∗ ∂y∗ ∂x∗ Re  ∂x∗2 ∂y∗2 

∗ ∗ ∗ ∗ 2 ∗ 2 ∗ ∂v ∗ ∂v ∗ ∂v ∂P 1 ∂ v ∂ v + u + v = − + + ; ∂t∗ ∂x∗ ∂y∗ ∂y∗ Re  ∂x∗2 ∂y∗2  ∂u∗ ∂v∗ + =0 , ∂x∗ ∂y∗ where Re = L U/ν is Reynolds number. Re is the only parameter of the problem and describes the contribution of • the viscosity Re ≫ 1 ⇒ viscosity is “small” Re ≪ 1 ⇒ viscosity is “large” • •

1. BOUNDARY LAYER 5 Similarity Laws ◦1 ◦2



















































































































































U




































U

















































































































































































































































































































































































L L

∗ ∗ ∗ ∗ (x, y)= L1 (x , y ); (x, y)= L2 (x , y ); ∗ ∗ ∗ ∗ (u, v)= U1 (u ,v ); (u, v)= U2 (u ,v );

*




































··· U =1









































































···



















































































































































L*=1

∗ ∗ ∗ ∗ 2 ∗ 2 ∗ ∂u ∗ ∂u ∗ ∂u ∂P 1 ∂ u ∂ u ∗ + u ∗ + v ∗ = ∗ + ∗2 + ∗2 ; ∂t ∂x ∂y − ∂x Re1,2 „ ∂x ∂y «

··· If Re1 = Re2 the equations describing the flow are identical and the flows • described in nondimensional variable are identical

1. BOUNDARY LAYER 6

Similarity Laws

Example 2: Fluid flow around an elastic structure



Motion of a 2D structure with 1 DoF: 2 Y( t ) Y¨ + ω0 Y = F (t)/m U F( t )






















The force F (t) is given by solution of fluids




































































equations coupled with structural equa-













































L tions

a Parameters: . Structure: ω0– natural frequency; m– mass ; Fluid: U; L; ν; ρ . ∗ ∗ 2 2 ∗ Nondimensional variables: t = t /ω0; Y = L Y ; F = ρU L F ∗ ∗ ∗ ∗ 2 ∗ (x, y)= L (x , y ); (u, v)= U (u ,v ); P = ρU P Change of variables: 2 ∗ ∗ ∗ 2 2 ∗ 2 d Y ∂u 2 ∂u ∂u U ∂u ∂ u U ∂ u Y¨ = ω0 L ; = U ω0 ; = ; = ; d t∗2 ∂t ∂t∗ ∂x L ∂x∗ ∂x2 L ∂x∗2 ···

aOr generalised mass, e.g. moment of inertia for torsion (not units of mass!)

1. BOUNDARY LAYER 7 Similarity Laws Equations become: Structure: 2 ∗ 3 2 d Y ∗ ρL U ∗ ∗ 2 + Y = F (t ) ∗ m L ω0 Fluid: dt   ∗ ∗ ∗ ∗ 2 ∗ 2 ∗ L ω0 ∂u ∗ ∂u ∗ ∂u ∂P ν ∂ u ∂ u + u + v = − + + ; U ∂t∗ ∂x∗ ∂y∗ ∂x∗ L U  ∂x∗2 ∂y∗2  ··· Similarity parameters: LU Re = – Reynolds number; viscous effects ν L ω0 Sr = – Strouhal number; unsteady effects U

∗ m m = – normalised mass; interaction ρL3 • Dynamically similar models

1. BOUNDARY LAYER 8

Similarity Laws Example 3: Flows with a free surface Parameters: H; D; h – geometrical U; ν; ρ g – gravity





h U











H





D





Nondimensional variables:





∗ ∗ ∗ ∗
































(x, y)= H (x , y ); (u, v)= U (u ,v )
































2 ∗ ∗ ∗ P = ρU P or P = ρgHP ; t =(H/U) t Nondimensional equations:

∗ ∗ ∗ ∗ 2 2 ∗ 2 ∗ ∂v ∗ ∂v ∗ ∂v ∂P √g H ν ∂ v ∂ v ; +u +v = + + + ; ··· ∂t∗ ∂x∗ ∂y∗ − ∂y∗ „ U « HU „ ∂x∗2 ∂y∗2 « ···

Re = H U/ν – Reynolds number; viscous effects Fr = U/√g H – Froude number; gravity effects D/h; h/H – length ratios; geometry

1. BOUNDARY LAYER 9