Some Basic Plane Potential Flows Uniform Stream in the x Direction A uniform stream V = iu, as in the Fig. (Solid lines are streamlines and dashed lines are potential lines), possesses both a stream function and a velocity potential, which may be found as follows: Integrating each expression and discarding the constants of integration, which do not affect the flow velocities, we get The streamlines are horizontal straight lines (y = const),and the potential lines are vertical (x = const), i.e., orthogonal to the streamlines, as expected. 34
Line Source or Sink at the Origin Consider a fluid flowing radially outward from a line along z -axisatatotalrateq uniformly along its length b. Looking at the xy plane, we would see a cylindrical radial outflow or line source, as sketched in the Fig. In this case, plane polar coordinates are appropriate, and there is no circumferential velocity. At any radius r, the velocity is Integrating and discarding the constants of integration, we obtain the proper functions for this simple radial flow: where m = Q/(2πb) is a constant. t Thus, the streamlines are radial spokes (constant θ), and the potential lines are circles (constant r). 35
Line Source or Sink at the Origin If m is positive, the flow is radially outward and is considered to be a source flow. If m is negative, the flow is radially inward and is considered to be a sink flow. It should be noted that at the origin where r = 0, the velocity becomes infinite, which is physically impossible. Thus, sources and sinks do not really exists in real flow fields, and the line representing the source or sink is a mathematical singularity i in the flow field. However, some real flows can be approximated at points away from the origin by using sources or sinks. 36
Line Irrotational Vortex A (two-dimensional) line vortex is a purely circulating steady motion with and. Here, the streamlines are concentric circles, c i.e. the velocity potential and stream function for source are interchanged. Thus, let ; where K is a constant called the strength of the vortex. Thisis sometimes called a free vortex, for which the velocity components may be found as: 37
Line Irrotational Vortex The tangential velocity, So, the tangential velocity varies inversely with the distance from the origin, with a singularity at r = 0 (where the velocity becomes infinite). It may seem strange that the vortex motion is irrotational (and it is since the flow field is described by a velocity potential). It may be noted the rotation refers to the orientation of a fluid particle and not the path followed by the element. The "bathtub vortex," formed when water drains through a bottom hole in a tank, is a good approximation to the free-vortex pattern. 38
Laplace Equation in φ and ψ If viscous effects are neglected, low-speed flows are irrotational (i.e. curl V = 0), and the velocity potential φ exists, such that The continuity equation, reduces to Laplace's eq. for φ On the other hand, if a flow is described by only two coordinates, the stream function ψ also exists as an alternate approach. For plane incompressible flow in xy coordinates, the velocity components are Thecondition of irrotationality ti reduces to Laplace's equation for ψ : 39
Laplace Equation in φ and ψ In Polar coordinate system, both the velocity components and the differential relations for φ and ψ are as follows: Laplace's equation takes the form Exactly the same equation holds for the polar-coordinate form of ψ (r, θ). For potential flows with no free surface, the governing equations (i.e., the Laplace's equation) contain no parameters, nor do the boundary conditions. There is no Reynolds, Froude, or Mach number to complicate the dynamic similarity. Inviscid id flows are kinematicallyi similar il without additional i parameters. 40
Superposition of Plane Flow Solutions Each of the three elementary flow patterns already discussed is an incompressible irrotational flow and therefore satisfies both plane "potential flow" equations, and Since these are linear partial differential equations, anysumof such basic solutions is also a solution. Some of these composite solutions are quite interesting and useful. Source plus an equal line sink Consider a source +m at (x, y) =(-a, 0), combined with a sink of equal strength -m, placed at (+a, 0), as in the Fig. 4.13. The resulting stream function is simply the sum of the two. Similarly, the composite velocity potential is 41
Superposition of Plane Flow Solutions Figure: Potential flow due to a line source plus an equal line sink. Solid lines are streamlines; dashed dlines are potential ti llines. These lines are plotted in the above Fig. and are seen to be two families of orthogonal circles, with the streamlines passing through the source and sink and the potential lines encircling them. 42
Superposition of Plane Flow Solutions Sink Plus a Vortex at the Origin An interesting flow pattern occurs by superposition of a sink and a vortex, both centered at the origin. The composite stream function and velocity potential are When plotted, these form two orthogonal families of logarithmic spirals, as shown in the Fig. This is a fairly realistic simulation of a tornado (where the sink flow moves up the z-axis into the atmosphere). At the center of a real (viscous) vortex, where Eq. (4.134) predicts infinite velocity, the actual circulating flow is highly rotational and approximates solid-body rotation 43
Superposition of Plane Flow Solutions Uniform Stream + a Sink at the Origin: The Rankine Half Body If a uniform x-directed stream is superimposed against an isolated source, a half-body shape appears. If a source is at the origin, the composite stream function and velocity potential are, (in polar coordinates) If the streamlines are plotted a curved, roughly elliptical, half-body shape appears, which separates the source flow from the stream flow. 44
Superposition of Plane Flow Solutions The body shape, known as Rankine half body, isformedbythe particular streamlines ψ = ± πm. The hlf half-width of the bd body far downstream is πm/u. The upper surface may be plotted from the relation It is not a true ellipse. The nose of the body, which is a "stagnation" point where V = 0, stands at (x, y) =(-a, 0),wherea = m/u. The streamline ψ = 0 also crosses this point. 45
Superposition of Plane Flow Solutions Rankine half body If the origin contains a source, a plane half-body is formed with its nose to the left, as in Fig. on the left. If the origin is a sink, m <0, the half-body nose is to the right, as in Fig. on the right. In either case the stagnation point is at a position away from the origin. 46
Rankine half body An offshore power plant cooling-water intake sucks in 1500 ft 3 /s in water 30 ft deep. If the tidal velocity approaching the intake is 0.7 ft/s, (a) how far downstream does the intake effect extend and (b) how much width L of tidal flow is entrained into the intake? The cooling water intake may be assumed to beasink.thesinkstrengthm is related to the volume flow Q and the depth b into the paper Therefore the desired lengths a and L are 47
Doublet An important basic potential flow to be considered is one that is formed by combining a source and sink in a special way. Consider the equal strength, source-sink pair of figure 6.21. The combined stream function for the pair is which can be rewritten as (6.92) From Fig. 6.21 it follows thatt and 48
Doublet These results substituted into Eq. 692 give so that For small values of a (6.94) since the tetangent t of an angle ageapproaches the tevalue of the teage angle for small angles. The so-called doublet is formed by letting the source and sink approach one another (a 0) while increasing the strength m (m ) so that the product 2ma remain constant. In this case, Eq.6.94 reduces to where K a constant equal to where K, a constant equal to 2ma, is called the strength of the doublet. 49
Doublet The velocity potential for the doublet is Plots of lines of constant ψ reveal that the streamlines for a doublet are circles through origin tangent to the x axis as shown in Fig. 6.22. Just as sources and sinks are not physically realistic entities, neither are doublets. However, the doublet when combined with other basic potential flows provides a useful representation of some flow fields of practical interest. For example, the combination of a uniform flow and a doublet can be used to represent the flow around a circular cylinder. 50
Flow Around a Circular Cylinder A uniform flow in the positive x direction combined with a doublet can be used to represent flow around a circular cylinder. This combination gives for the stream function and for the velocity potential In order for the stream function to represent flow around a circular cylinder it is necessary that ψ = constant for r = a, wherea is the radius of the cylinder. Since Eq.6.103 can be written as it follows that ψ =0forr = a if which indicates that the doublet strength, K must be equal to Ua 2. Thus, the stream function for flow around a circular cylinder can be expressed as 51
Flow Around a Circular Cylinder and the corresponding velocity potential is A sketch of the streamlines for this flow field is shown in Fig 6.24 The velocity components can be obtained from either the stream function ot the velocity potential as and On the surface of the cylinder (r = a) it follows that and This gives maximum velocity at the top and bottom of the cylinder (θ =±π/2) and has a magnitude of twice the upstream velocity. 52
Flow Around a Circular Cylinder The pressure distribution on the cylinder surface is obtained from the Bernoulli equation written from a point far from the cylinder where the pressure is p 0 and the velocity is U so that where p S is the surface pressure. Elevation changes are neglected. Since the surface pressure can be expressede edas A comparison of this theoretical pressure distribution expressed in dimensionless form is with a typical measured distribution is shown in Fig. 6.25. 53
Flow Around a Circular Cylinder This figure clearly reveals that there is approximate agreement between the potential flow and the experimental results only on the upstream part of the cylinder. Because of the viscous boundary layer that develops on the cylinder, the main flow separates from the surface of the cylinder, leading to the large difference between the theoretical, frictionless fluid solution and the experimental results on the downstream side of the cylinder. The resultant force (per unit length) developed on the cylinder can be determined by integrating the pressure over the surface. 54
Flow Around a Circular Cylinder From Fig. 6.26 it can be seen that and where F x is the drag (force parallel to direction of the uniform flow) and F y is the lift (force perpendicular to the direction of the uniform flow). Substitution for p S into these two equations, and subsequent integration, reveals that F x =0andF y =0. These results indicate that both the drag and lift as predicted by potential theory for a fixed cylinder in a uniform stream are zero. However, we know from experiments that there is a significant drag developed on a cylinder when it is placed in a moving fluid. This discrepancy is known as d'alembert's paradox. 55
Circulation Circulation is a mathematical concept commonly associated with vortex motion. The circulation, Γ, is defined as the line integral of the tangential component of the velocity taken around a closed curve in the flow field. In equation form, Γ can be expressed as (6.89) Where the integration is taken around a closed curve, C, in the counterclockwise direction, and ds is a differential length along the curve as illustrated in Fig. 6.19. For an irrotational flow, and, therefore, so that 56
Circulation This result indicates that for an irrotational flow the circulation will generally be zero. However, if there are singularities enclosed within the curve the circulation may not be zero. For example, for the free vortex with the circulation around the circular path of radius r shown in Fig. 6.20 is which shows that the circulation is non zero and the constant. However, the circulation around any path that does not include the singular point at the origin will be zero. 57
Circulation The velocity potential and stream function for the free vortex are commonly expressed in terms of the circulation as and The concept of circulation is often useful when evaluating the forces developed on immersed in moving fluids. An interesting potential flow can be developed by adding a free vortex to the stream function or velocity potential for the flow around a cylinder. In this case and where Γ is the circulation. 58
Flow Around a Rotating Circular Cylinder The circle r = a will be a streamline (and thus can be replaced with a solid cylinder). However, the tangential velocity,, on the surface of the cylinder (r = a) now becomes (6.114) This type of flow field could be approximately created by placing a rotating cylinder in a uniform stream. Because of the presence of viscosity in any real fluid, the fluid in contact with the rotating cylinder would rotate with the same velocity as the cylinder, and the resulting flow field would resemble thatt developed d by the combination of auniform flow past acylinder and a free vortex. Putting in Eq. 6.114, wecandetermine dt the location of stagnation point on the surface of the cylinder as 59
Flow Around a Rotating Circular Cylinder If - i.e., the stagnation points occur at the front and rear of the cylinder as are shown in Fig. 6.27a. (6.115) For, the stagnation points will occur at some other location on the surface as illustrated in 6.27b, c. If the absolute value of the parameter exceeds 1, the stagnation point is located away from the cylinder as shown in Fig. 6.27d. 60
Flow Around a Rotating Circular Cylinder The force per unit length developed on the cylinder can be obtained by integrating the differential pressure forces around the circumference. For the cylinder with circulation, the surface pressure p s, is obtained from the Bernoulli equation (with the surface velocity given by Eq. 6.114) Or 6.116) The drag and lift may be determined as follows Drag or Lift That is for the rotating cylinder no drag is developed. However, lift is developed equal to the product of ρ, U,andΓ. 61 or
Flow Around a Rotating Circular Cylinder The negative sign in the expression of lift means that if U is positive (in the positive x direction) and Γ is positive (a free vortex with counterclockwise rotation), the direction of the lift, F y is downward. Of course, if the cylinder is rotated in the clockwise direction (Γ <0) the direction of F y would be upward. It is this force acting in a direction perpendicular to the direction of the approach velocity that causes baseballs and golf balls to curve when they spin as they are propelled through the air. The development of this lift on rotating bodies is called the Magnus effect. The generalized equation relating lift to the fluid density, velocity, and circulation is called the Kutta-Joukowski Law and is commonly used to determine the lift on airfoils 62