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2 016 Hydrodynamics Reading 4,Laplace Equation, The velocity must still satisfy the conservation of mass equation We can substitute in. the relationship between potential and velocity and arrive at the Laplace Equation which. we will revisit in our discussion on linear waves,u v w 0 4 2. x 2 y 2 z 2,LaplaceEquation 2 0, For your reference given below is the Laplace equation in different coordinate systems. Cartesian cylindrical and spherical,Cartesian Coordinates x y z. V ui v j wk i j k,Cylindrical Coordinates,r 2 x 2 y 2 tan 1 y x.
V ur e r u e u z e z e r e e z,version 1 0 updated 9 22 2005 2 2005 A Techet. 2 016 Hydrodynamics Reading 4,Spherical Coordinates r. r 2 x 2 y 2 z 2 cos 1 x r or x r cos tan 1 z y,V ur e r u e u e e r e e. 1 r r 2 sin,2r4 3 r 2 sin 2 2,Potential Lines, Lines of constant are called potential lines of the flow In two dimensions. Since d 0 along a potential line we have,Recall that streamlines are lines.
everywhere tangent to the velocity so potential, lines are perpendicular to the streamlines For inviscid and irrotational flow is indeed. quite pleasant to use potential function to represent the velocity field as it reduced. the problem from having three unknowns u v w to only one unknown. As a point to note here many texts use stream function instead of potential function as it. is slightly more intuitive to consider a line that is everywhere tangent to the velocity. Streamline function is represented by Lines of constant are perpendicular to lines. of constant except at a stagnation point,version 1 0 updated 9 22 2005 3 2005 A Techet. 2 016 Hydrodynamics Reading 4, Luckily and are related mathematically through the velocity components. Equations 4 5 and 4 6 are known as the Cauchy Riemann equations which appear in. complex variable math such as 18 075,Bernoulli Equation. The Bernoulli equation is the most widely used equation in fluid mechanics and assumes. frictionless flow with no work or heat transfer However flow may or may not be. irrotational When flow is irrotational it reduces nicely using the potential function in. place of the velocity vector The potential function can be substituted into equation 3 32. resulting in the unsteady Bernoulli Equation,1 2 p g z 0.
V 2 p gz 0 4 8,V 2 p gz c t,UnsteadyBernoulli 4 9,version 1 0 updated 9 22 2005 4 2005 A Techet. 2 016 Hydrodynamics Reading 4,Potential Stream Function. Definition V V,Continuity 2 0 Automatically Satisfied. Irrotationality Automatically Satisfied v v v,In 2D w 0 0. 2 0 for continuity z 2 0 for,irrotationality, Cauchy Riemann Equations for and from complex analysis.
i where is real part and is the imaginary part,Cartesian x y. For irrotational flow use,For incompressible flow use. For incompressible and irrotational flow use and,version 1 0 updated 9 22 2005 5 2005 A Techet. 2 016 Hydrodynamics Reading 4,Potential flows, Potential functions and stream functions can be defined for various simple flows. These potential functions can also be superimposed with other potential functions to. create more complex flows,Uniform Free Stream Flow 1D.
V Ui 0 j 0 k 4 10, We can integrate these expressions ignoring the constant of integration which ultimately. does not affect the velocity field resulting in and. Ux and Uy 4 13, Therefore we see that streamlines are horizontal straight lines for all values of y tangent. everywhere to the velocity and that equipotential lines are vertical straight lines. perpendicular to the streamlines and the velocity as anticipated. 2D Uniform Flow V U V 0 Ux Vy Uy Vx, 3D Uniform Flow V U V W Ux Vy Wz no stream function in 3D. version 1 0 updated 9 22 2005 6 2005 A Techet,2 016 Hydrodynamics Reading 4. Line Source or Sink, Consider the z axis into the page as a porous hose with fluid radiating outwards or.
being drawn in through the pores Fluid is flowing at a rate Q positive or outwards for a. source negative or inwards for a sink for the entire length of hose b For simplicity take. a unit length into the page b 1 essentially considering this as 2D flow. Polar coordinates come in quite handy here The source is located at the origin of the. coordinate system From the sketch above you can see that there is no circumferential. velocity but only radial velocity Thus the velocity vector is. V ur e r u e u z e z ur e r 0 e 0 e z 4 14,Integrating the velocity we can solve for and. m ln r and m 4 17, where m Note that satisfies the Laplace equation except at the origin. r x 2 y 2 0 so we consider the origin a singularity mathematically speaking and. exclude it from the flow, The net outward volume flux can be found by integrating in a closed contour. around the origin of the source sink,V n dS V dS,version 1 0 updated 9 22 2005 7 2005 A Techet. 2 016 Hydrodynamics Reading 4,Irrotational Vortex Free Vortex.
A free or potential vortex is a flow with circular paths around a central point such that the. velocity distribution still satisfies the irrotational condition i e the fluid particles do not. themselves rotate but instead simply move on a circular path See figure 2. Figure 2 Potential vortex with flow in circular patterns around the center. Here there is no radial velocity and the individual particles do not rotate. about their own centers, It is easier to consider a cylindrical coordinate system than a Cartesian coordinate system. with velocity vector V ur u u z when discussing point vortices in a local reference. frame For a 2D vortex u z 0 Referring to figure 2 it is clear that there is also no. radial velocity Thus,V ur e r u e u z e z 0 e r u e 0 e z 4 18. Let us derive u Since the flow is considered irrotational all components of the. vorticity vector must be zero The vorticity in cylindrical coordinates is. 1 u z u u u 1 ru 1 ur,V e r r z e ez 0 4 21,r z z r r r r. version 1 0 updated 9 22 2005 8 2005 A Techet,2 016 Hydrodynamics Reading 4. ur2 u x2 u y2 4 22,u u x cos 4 23,u z 0 for 2D flow.
Since the vortex is 2D the z component of velocity and all derivatives with respect to z. are zero Thus to satisfy irrotationality for a 2D potential vortex we are only left with the. z component of vorticity e z, Since the vortex is axially symmetric all derivatives with respect must be zero Thus. From this equation it follows that ru must be a constant and the velocity distribution for. a potential vortex is,u ur 0 u z 0 4 26, By convention we set the constant equal to where is the circulation Therefore. Figure 3 Plot of velocity as a function of radius from the vortex center. At the core of the potential vortex the velocity blows up to infinity and is. thus considered a singularity,version 1 0 updated 9 22 2005 9 2005 A Techet. 2 016 Hydrodynamics Reading 4, You will notice see figure 3 that the velocity at the center of the vortex goes to infinity. as r 0 indicating that the potential vortex core represents a singularity point This is. not true in a real or viscous fluid Viscosity prevents the fluid velocity from becoming. infinite at the vortex core and causes the core rotate as a solid body The flow in this core. region is no longer considered irrotational Outside of the viscous core potential flow can. be considered acceptable,Integrating the velocity we can solve for and.
K and K ln r 4 28, where K is the strength of the vortex By convention we consider a vortex in terms of its. circulation where 2 K is positive in the clockwise direction and represents the. strength of the vortex such that,and ln r 4 29, Note that using the potential or stream function we can confirm that the velocity field. resulting from these functions has no radial component and only a circumferential. velocity component, The circulation can be found mathematically as the line integral of the tangential. component of velocity taken about a closed curve C in the flow field The equation for. circulation is expressed as, where the integral is taken in a counterclockwise direction about the contour C and. ds is a differential length along the contour No singularities can lie directly on the. contour The origin center of the potential vortex is considered as a singularity point in. the flow since the velocity goes to infinity at this point If the contour encircles the. potential vortex origin the circulation will be non zero If the contour does not encircle. any singularities however the circulation will be zero. version 1 0 updated 9 22 2005 10 2005 A Techet,2 016 Hydrodynamics Reading 4.
To determine the velocity at some point P away from a point vortex figure 4 we need. to first know the velocity field due to the individual vortex in the reference frame of the. vortex Equation Error Reference source not found can be used to determine the. tangential velocity at some distance ro from the vortex It was given up front that ur 0. everywhere, Since the velocity at some distance ro from the body is constant on a circle centered on. the vortex origin the angle o is not crucial for determining the magnitude of the. tangential velocity It is necessary however to know o in order to resolve the direction. of the velocity vector at point P The velocity vector can then be transformed into. Cartesian coordinates at point P using equations 4 22 and 4 23. Figure 4 Velocity vector at point P due to a potential vortex with strength. located some distance ro away,version 1 0 updated 9 22 2005 11 2005 A Techet. 2 016 Hydrodynamics Reading 4,Linear Superposition. All three of the simple potential functions presented above satisfy the Laplace equation. Since Laplace equation is a linear equation we are able to superimpose two potential. functions together to describe a complex flow field Laplace s equation is. x 2 y 2 z 2, Let 1 2 where 2 1 0 and 2 2 0 Laplace s equation for the total potential is. 2 2 1 2 2 1 2 2 1 2,x 2 y 2 z 2,2 2 2 2 2 2,2 21 22 21 22 21 22 4 32.
x x y y z z,2 2 2 2 2 2,2 21 21 21 22 22 22 4 33,x y z x y z. 2 2 1 2 2 0 0 0 4 34, Therefore the combined potential also satisfies continuity Laplace s Equation. Example Combined source and sink, Take a source strength m located at x y a 0 and a sink strength m located at. source sink m ln x a y 2 ln x a y 2, This is presented in cartesian coordinates for simplicity Recall r 2 x 2 y 2 so that. m ln r m ln x 2 y 2 m ln x 2 y 2, This is analogous to the electro potential patterns of a magnet with poles at a 0.
version 1 0 updated 9 22 2005 12 2005 A Techet,2 016 Hydrodynamics Reading 4. Example Multiple Point Vortices, Since we are able to represent a vortex with a simple potential velocity function we can. readily investigate the effect of multiple vortices in close proximity to each other This. can be done simply by a linear superposition of potential functions Take for example two. vortices with circulation 1 and 2 placed at a along the x axis see figure 5 The. velocity at point P can be found as the vector sum of the two velocity components V1 and. V2 corresponding to the velocity generated independently at point P by vortex 1 and. vortex 2 respectively, Figure 5 Formulation of the combined velocity field from two vortices in. close proximity to each other Vortex 1 is located at point x y a 0. and vortex 2 at point x y a 0, The total velocity potential function is simply a sum of the potentials for the two. individual vortices,T v1 v 2 1 2 1 2 2, with 1 and 2 taken as shown in figure 5 One vortex in close proximity to another vortex.
tends to induce a velocity on its neighbor causing the free vortex to move. White Fluid Mechanics 4th ed We can treat external flows around bodies as invicid i e frictionless and irrotational i e the fluid particles are not rotating This is because the viscous effects are limited to a thin layer next to the body called the boundary layer In graduate classes like 2 25