CEE 262A H YDRODYNAMICS Lecture 4 Kinematics Part II 1

Stream Function (a) We will show that if conservation of mass (continuity) is: Then for an incompressible or slightly compressible fluid A

(b) Iff Then becomes A B (c) A function can be defined such that Check from B

(d) Whenever can be defined At any instant In addition, from before along streamlines C D From & along a streamline DC

(e) Finally, it can be shown (see Kundu Sec. 3.13) that y x The volume flowrate between streamlines is numerically equal to the difference in their values. (f) See appendix in text for definition of  in different coordinate systems

E.g. flow around a cylinder: Uniform part Deviation caused by cylinder Nx = 60; % Number of points in x-direction Nz = 60; % Number of points in z-direction xmin = -3; % Minimum x-coordinate xmax = 3; % Maximum x-coordinate zmin = -3; % Minimum z-coordinate zmax = 3; % Maximum z-coordinate [x,z]=meshgrid(linspace(xmin,xmax,Nx),linspace(zmin,zmax, Nz)); % Streamfunction psi=z-z./(x.^2+z.^2); % Contour values cv = [-4:.2:4]; % Contour plot figure(1) contour(x,z,psi,cv,'k-') xlabel('x'); ylabel('z'); axis image; streamlines.m

Vortex flows (a) Vorticity is the curl of the velocity field (b) Vorticity is also the circulation per unit area From Stokes Theorem - “Component of vorticity through a surface A bounded by C equals the line integral of the velocity around C.” - If we define circulation Circulation = Total amount of vorticity ┴ to a given area; or flux of vorticity through a given area. Then

If a body consists of elements rotating at a rate  0 about its origin and if each element rotates at  0 about its own origin, then the body is in solid body rotation. Solid-body rotation

(i)Vorticity normal to page (see Kundu appendix on vorticity in plane polar coordinates): (ii) (iii) Stokes?

Point or irrotational Vortex

What about circulation? If we compute average vorticity for some r, we get: This is a vortex singularity = an infinitely strong bit of vorticity located on an infinitely small area Can it exist in reality? No, but….. a constant - independent of r

The difference between solid-body and irrotational vortices: Irrotational vortex: There is a lot of shear-strain, but the average rotation rate of two intersecting segements is zero. Solid-body rotation: The parcel rotates at the angular velocity given by that of the solid-body rotation. irrot_vs_solidbody.m

It provides a good model for things that happen in nature where the vorticity is quite concentrated. e.g. tidal inlet. What happens if we place two point vortices of opposite signs but the same strength, a distance 2a apart? Each will be characterized by circulation  0, so that the velocity either one “induces” a distance r away is  0 /2  r 2a +  0 -  0 This means that the right one make the left ascend at speed  0 /4  a, while the left one has the same effect on the right... Point vortex

Vortices "passive" Lagrangian tracers vortexpair.m

(Turned on its side – i.e. going to left or right, and held in place)

Flows without vorticity are known as “irrotational”, or potential. For these, we can write  is known as the velocity potential. This relation comes about because (as we showed earlier) If the flow is also incompressible, The physical structure of irrotational flows is often determined by the geometry (which imposes b.c.s on the Laplace equation). In 2D, we can also use the streamfunction which gives (also)

Why are streamlines and equipotentials always perpendicular to each other?