Lecture 16 D(drag)xF(lift)yF0U - Marine Hydrodynamics Lecture 16 4.1.4 Vortex Shedding and Vortex Induced Vibrations Consider a steady flow Uo over a bluff body with diameter D. We would expect the average forces to be: FyFxtFHowever, the measured oscillatory forces are: Average Average tFFxFy• The measured drag Fx is found to oscillate about a non-zero mean value with frequency 2f. • The measured lift Fy is found to oscillate about a zero mean value with frequency f. • f = ω/2π is the frequency of vortex shedding or Strouhal frequency. 1 2.20 - Marine Hydrodynamics, Spring 20052.20UoDVon Karman StreetFyFxReason: Flow separation leads to vortex shedding. The vortices are shed in a staggered array, within an unsteady non-symmetric wake called von Karman Street. The frequency of vortex shedding is the Strouhal frequency and is a function of Uo, D, and ν. i) Strouhal Number We define the (dimensionless) Strouhal number S ≡ Strouhal frequency ����f D U0 . The Strouhal number S has a regime dependence on the Re number S = S(Re). 1051061070.220.3S(Re)ReFor a cylinder: Laminar flow S ∼ 0.22 • Turbulent flow S ∼ 0.3 • ii) Drag and Lift The drag and lift coefficients CD and CL are functions of the correlation length. For ‘∞’ correlation length: If the cylinder is fixed, CL ∼ O(1) comparable to CD. • • If the cylinder is free to move, as the Strouhal frequency fS approaches one of the cylinder’s natural frequencies fn, ‘lock-in’ occurs. Therefore, if one natural frequency is close to the Strouhal Frequency fn ∼ fS , we have large amplitude motions Vortex Induced Vibration (VIV).⇒ 2LbDUo4.2 Drag on a Very Streamlined Body UL ReL ≡ ν Cf ≡ D 1 2 ρU2 (Lb)����S=wetted area one side of plate Cf = Cf (ReL , L/b) Cf0.01Laminar105Re0.001Turbulent106= ν ∂u �Unlike a bluff body, Cf is a strong function of ReL since D is proportional to ν �τ ∂y . See an example of Cf versus ReL for a flat plate in the figure below. Skin friction coefficient as a function of the Re for a flat plate • ReL depends on plate smoothness, ambient turbulence, . . . In general, Cf ’s are much smaller than CD’s (Cf /CD ∼ O(0.1) to O(0.01)). Therefore, • designing streamlined bodies allows minimal separation and smaller form drag at the expense of friction drag. • In general, for streamlined bodies CTotal Drag is a combination of CD (Re) and Cf (ReL ), 1 � �and the total drag is D = 2 ρU2 CD S + Cf Aw , where CD has a regime frontal area wetted area dependence on Re and Cf is a continuous function ReL . 34.3 Known Solutions of the Navier-Stokes Equations 4.3.1 Boundary Value Problem Navier-Stokes’: • ∂�v 1 1 ∂t + (�v · �) �v = −ρ�p + ν� 2�v + ρf�Conservation of mass: • � · �v = 0 • Boundary conditions on solid boundaries “no-slip”: �v = U�Equations very difficult to solve, analytic solution only for a few very special cases (usually when � v = 0. . . ) v · ��44.3.2 Steady Laminar Flow Between 2 Long Parallel Plates: Plane Couette Flow yhzxUSteady, viscous, incompressible flow between two infinite plates. The flow is driven by a pressure gradient in x and/or motion of the upper plate with velocity U parallel to the x-axis. Neglect gravity. Assumptions Governing Equations Boundary Conditions i. Steady Flow: ∂ ∂t = 0 ii. (x, z) >> h: ∂�v ∂x = ∂�v ∂z = 0 iii. Pressure: independent of z Continuity: ∂u ∂x + ∂v ∂y + ∂w ∂z = 0 NS: ∂�v ∂t + �v · ��v = −1 ρ �p + ν�2�v �v = (0, 0, 0) on y = 0 �v = (0, 0, 0) on y = h Continuity ∂u ∂v ∂w ∂v + + = 0 = 0 v = v(x, z) v = 0 (1)∂x ∂y ∂z⇒ ∂y ⇒ ⇒���� ����BC: v(x,↑0,z)=0 =0, from assumption ii 5����Momentum x ∂u ∂u ∂u ∂u 1 ∂p � ∂2u ∂2u ∂2u � +u + v + w = − + ν + + ∂t ∂x����∂y ∂z ρ ∂x ∂x2 ∂y2 ∂z2 ⇒=0, (1)����=0, ����ii ���� ����ii ����i =0, =0, ii =0, =0, ii ∂2u 1 ∂p ν = (2)∂y2 ρ ∂x Momentum y ∂v 1 ∂p +� v2 v ∂tv · � =0����, = −ρ ∂y + ν� ����(1) ⇒ (1) =0, =0, i ∂p ∂p dp= 0 p = p(x) and = (3)∂y ⇒∂x dx↑assumption iii Momentum z ∂w ∂w ∂u ∂w 1 ∂p � ∂2w ∂2w ∂2w � +u + v + w = − +ν + + ∂t ∂x����∂y ∂z ρ ∂z ∂x2 ∂y2 ∂z2 ⇒=0, (1)���� ����ii ���� ����iii ���� �=0��, �ii=0, i =0, =0, ii =0, =0, ii ∂2w = 0 w = ay + b w = 0 (4)∂y2 ⇒ ⇒↑w(x,0,z)=0 w(x,h,z)=0 From Equations (1), (4) ∂u du �v = (u, 0, 0). Also u = u(y) and = (5)⇒∂y dy↑assumption ii From Equations (2), (3, and (5) d2u 1 d2p 1 � dp � 21 � dp �y dy2 = ρν dx2 ⇒ u = −2µ −dx y +C1y+C2 ⇒ u = − (h − y)y + U 2µ dx h µ=↑ρν u(x,0↑,z)=0 w(x,h,z)=U 6y)( yu0)(==Uhu0)0(=upp0)( >−dxdpprofileParabolichy)( yuUhu=)(0)0(=u0)( =−dxdpprofileLinearhUSpecial cases for Couette flow • 1 dp dpPx−Px+Lu(y) = 2µ (h − y)y(−dx ) + U hy , where (−dx ) = L dp dpI. U = 0, � − � > 0 II. U = 0� , � − � = 0dx dx Velocity•u(y) = 1 dp ) u(y) = U y 2µ (h − y)y(−dx h Max velocity•dpumax = u(h/2) = 8hµ 2 (−dx ) umax = U Volume flow rate • h3 dpQ = �0 h u(y)dy = 8µ (−dx ) Q = h U2 Average velocity•Qh2 dpu¯ = h = 6µ (−dx ) u¯ = U 2 Viscous stress on bottom plate (skin friction)•du ���Uτw = µ du ���= h dp � > 0 τw = µ dy = µ hdy 2 � − dx y=0 y=0 70,0)(,0 >>−> GdxdpUwτUhu=)(hU0,0)(,0 <<−> GdxdpUflowbackUhu=)(UwτIII. U �= 0, � − dp dx � �= 0 � − dp dx � > 0 � − dp dx � < 0 •Viscous stress on bottom plate (skin friction) τw = h 2 � − dp dx � + µ U h τw < = >0 when (− dp dx) < = > − 2µU h2 , in which case the flow is � attached insipient separated 80,0)(,0 >>−> GdxdpUUhu=)(hU0,0)(,0 <<−> GdxdpU01,0)(,0 =⇒−=<−>wGdxdpUτUhu=)(U UUhu=)(flowbackdpFor the general case of U = 0 and �� − � = 0, �dx τw = h� − dp � + µU 2 dx h We define a Dimensionless Pressure Gradient G G ≡ h2 � − dp � 2µU dx such that G > 0 denotes a favorable pressure gradient • G < 0 denotes an adverse pressure