## Fluid MechanicsProperties of Fluids Fluid Statics Control Volume Analysis, Integral Methods Applications of Integral Methods Potential Flow Theory Examples of Potential Flow Dimensional Analysis Introduction to Boundary Layers Viscous Flow in Pipes |
## Elements of Potential FlowDifferential analysis of fluid motion is different from the analysis done in the previous section using control volumes. The focus now will be on a description of flow at any given point in the flow rather than on overall effects of the flow on a control volume. Point descriptions of flow are required to find the distribution of pressure on the surface of an aircraft wing or the distribution of velocities in a pipe for an air-conditioning system. This necessitates a more detailed knowledge of the flow field than that provided by the "integral approach" of the previous section. The laws of conservation of mass, momentum and energy are applied to produce differential forms of the governing equations (and hence the name "differential approach". When all components are included, including viscosity, this leads to what are called the Navier-Stokes Equations. In situations where viscous effects make up only a small proportion of the flow then simpler solutions which are reasonably accurate can be employed. Only inviscid, incompressible flows will be considered in this section. The solution is then based only on the equation for mass conservation. This will allow the calculation of velocity components. Pressure is then obtained from Bernoulli equation. The continuity equation will be derived. Then the concepts of stream function and velocity potential will be introduced. By analysing the kinematics of fluid motion, concepts of circulation and irrotationality can be demonstrated. Examples of simple flows such as a uniform flow, source and sink flow and vortex flow will be given. These flows can then be superimposed to produce solutions for more complex flows. Flow about a circular cylinder is analysed in some detail. ## Conservation of MassThe differential form of the equation for mass conservation can be found by considering a control volume at P(x,y,z) as shown in the figure below. The dimensions of the volume are dx, dy and dz and velocity components at point P will be u, v and w. Assuming that the mass flow rate is continuous across the volume we can calculate the mass flow rates at the various faces of the cell by a Taylor Series expansion. Accordingly, $$\text"MB"=(ρv-{∂ρv}/{∂y}.{dy}/2).dx.dz\text" , "\text"MT"=(ρv+{∂ρv}/{∂y}.{dy}/2).dx.dz$$ $$\text"MA"=(ρw-{∂ρw}/{∂z}.{dz}/2).dx.dy\text" , "\text"MF"=(ρw+{∂ρw}/{∂z}.{dz}/2).dx.dy$$ The net mass flow rate into the control volume as a consequence is given by, leading to, Applying the Reynolds transport theorem for mass will give, thus Cancelling out dx dy dz, we have, This equation is known as the Continuity Equation. Note that it is a very general equation with hardly any assumption except that density and velocities vary continuously across the element. ## Continuity Equation in Cylindrical CoordinatesThe continuity equation was derived above using cartesian coordinates. It is possible to use this system for all flows, but sometimes the equations may become cumbersome. Many flows which involve rotation or radial motion are best described in cylindrical coordinates. In this system, coordinates for point P are The flow rates are given by, $$\text"MB"=(ρv_θ-{∂ρv_r}/{∂r}{dr}/2).rdθ.dz\text" , "\text"MT"=(ρv_r+{∂ρv_r}/{∂r}{dr}/2).rdθ.dz$$ $$\text"MA"=(ρw-{∂ρw}/{∂z}{dz}/2).dr.rdθ\text" , "\text"MF"=(ρw+{∂ρw}/{∂z}{dz}/2).dr.rdθ$$ As a consequence the continuity equation becomes, ## Continuity Equation for steady flowFor a steady flow the time derivative vanishes. As a result continuity becomes $${∂ρv_r}/{∂r}+1/r{∂ρv_θ}/{∂θ}+{∂ρw}/{∂z} = 0$$ ## Continuity Equation for an Incompressible flowFor an incompressible flow density is a constant. Accordingly we have $${∂v_r}/{∂r}+1/r{∂v_θ}/{∂θ}+{∂w}/{∂z} = 0$$ As noticed for the control volume analysis the continuity equation for an incompressible flow is the same whether the flow is steady or unsteady. ## Velocity PotentialPhysically the driving force for fluid flow is pressure, however the introduction of a mathematical forcing function allows for a more standard equation and simpler solutions. Velocity potential $$v_r={∂φ}/{∂r}\text" , "v_θ=1/r{∂φ}/{∂θ}$$ substituting these definitions into the continuity equation leads to the governing potential flow equation. This is a standard version of the Laplace equation and many solutions are possible as will be seen later in this section. ## Stream-functionStream-function is amnother useful parameter for the study of fluid dynamics. It was arrived
at by the French mathematician Consider a two-dimensional incompressible flow for which the continuity equation is given by, Stream-function is a measure of volume flow rate between the streamlines of the flow, it is defined as a functional change at right angles to the flow direction, and as Velocity times Area is constant in incompressible flow, will predict the local velocities. A stream function, Substituting these expressions into the continuity equation will give, ## ψ, Stream-function is constant along a streamlineConsider a line given by after substituting for the gradients as velocities, -$v.dx+u.dy=0$ thus Flow is thus always tangential to the streamlines and there will be no flow across a stream line. ## Stream-function change (dψ) between two streamlines is proportional to the volumetric flowConsider
the volumetric flow through a small element of length ( The volumetric flow through the element is given by which indicates that the volumetric flow rate is proportional to the
difference between stream functions. If we now integrate the between
two stream lines ψ,
then,_{B}## Stream Function in Polar CoordinatesThe velocity components in polar coordinates are related to the stream function by, ## Kinematics of Fluid MotionA two-dimensional fluid particle, a square This complex deformation of the element can be split into four basic constituents -
Translation (movement of the centroid of the element) Linear Deformation (scaling of the size of the element) Rotation Angular Deformation (shearing of the element)
These deformations can be analysed separately to see their impact on the flow. ## TranslationTranslation is the type of motion where the element retains its shape. Its side
do not undergo any change in length and the four angles do remain
square. The square element Considering now a particle in the fluid element we can write down an expression
for its acceleration. At time where Consequently the change in velocity magnitude, The acceleration of the particle is obtained by dividing throughout by Now denoting the speed in x-direction, where dV
is usually denoted by _{p}/dtDV/Dt and is called material derivative or the
particle derivative or total derivative.
## Linear DeformationConsidering
the same element Now, the stretch in x-direction is given by distance The corresponding
change in the volume, Similarly the change in the volume of the element in the y-direction is given by, Neglecting the small change in volume due to Thus the rate of change of volume expressed as a fraction of the initial volume is given by, The left hand side is called the Volume Dilitation rate of the element. The right hand side of the equation is zero for all incompressible flows. Hence the volume dilitation rate for particles in potential flow will be zero. ## RotationConsidering the same element From the geometry, due to the rotation $$DD'=-{∂u}/{∂y}.dy.dt$$ Since angles $$dβ={DD'}/{AD}={DD'}/{dy}=-{∂u}/{∂y}.dt$$ Rotation or angular velocity of the element (about the z-axis) can be taken as the average of these two angular rates Combining the above information, Similarly, for rotation about the other axes, In most cases the term vorticity is used and is defined as twice
rotation rate, One group of flows has zero vorticity (i.e., rotation = 0 ). These are known as Irrotational Flows. For two dimensional, incompressible flows this is governed by the condition, All potential flows are irrotational. This can be shown by substituting the gradient definitions for velocity into the above equation for vorticity. It should be noted that the concept of Irrotationality applies to a fluid element in a given flow rather than to the flow itself. The main flow may contain a vortex where the streamlines are circles, but the individual elements of fluid may not rotate or distort making the flow irrotational. This is shown in the following figure where an irrotational flow about an aerofoil is shown. Note that even though the streamlines follow a path which seems to curve and indicate an overall flow rotation, the particles themselves translate but do not rotate. ## Angular deformationIf the particle is being sheared then the rate of shear strain for a two-dimensional flow this is given by the difference in rotation rate on each side. Shear stress for the element can thus defined by where ## CirculationCirculation is a measure of the total rotation induced by or contained in the flow field.
Consider any closed curve By
applying Green's theorem circulation can be shown to be the sum of
all vorticity within the area bounded by curve ## Occurrence of Irrotational or Rotational FlowsThe application of potential flow methods requires the flow to be irrotational. There are several cases where this can be applied and others where it cannot as the flow is rotational. A uniform flow is definitely irrotational, but this is a trivial flow and easy to predict. The region away from the surface of a solid body is also irrotational. However new the surface this is not the case. A velocity gradient ${∂u}/{∂y}$ is set up when a fluid flows along the surface due the shear. The velocity right on the body surface is zero and it increases quickly normal the the surface away from the body. This region is highly rotational and is called the boundary layer. It is usually very thin and only effects a small region near the surface. Also due the flow separation and upper and lower surface boundary layer mixing the wake region will contain fluid shear and is also NOT irrotational. It is worth noting that the flow is irrotational wherever the Bernoulli equation is valid. Most external flows are likely to be inviscid flow and hence irrotational. This is broadly true until the speed becomes excessive and shock waves occur in the flow. The region just behind a shock in a high speed flow has severe gradients of velocity causing large shear and rotation. ## Simple Examples of Plane Potential FlowsThe examples considered here are such that there are analytical
expressions for ## Potential Flow Equations in Cartesian CoordinatesThe velocity components are given by, The Laplacian is given by, ## Equations in Polar CoordinatesVelocity Components: The Laplacian is given by, In the following images, the flow is defined by blue streamlines, having constant stream function value and green equi-potentials, having constant velocity potential value. ## Uniform FlowThe simplest possible potential flow is a uniform flow, $V_{∞}$=constant $$u=V_{∞}={∂φ}/{∂x}\text" , "v={∂φ}/{∂y}=0$$ On integration, where The stream function for uniform flow can be easily calculated in a similar manner, and is given by, It is also possible to define velocity potential and stream function for uniform flows which incorporate angle of attack such that, ## Source or SinkConsidering a radial flow going away from the origin with a velocity, If or Velocity potential and stream function for this flow : It is easily verified that tangential velocity ($v_θ$)
is zero for this flow. The conservation of mass equation for the
source flow means that the Volumetric flow rate
(mass flow rate divided by density) is constant in a radial direction
and is equal to Note that the radial velocity r = 0. So the origin is a singularity of this flow. If m
is negative we have a flow which moves inward to the origin and is
called a Sink flow. This again has a singularity at the origin.## VortexAnother primary flow is one where radial velocity is zero and only flow in a circumferential direction occurs. This is Vortex flow. The velocity potential and stream function are given by, The velocity components are given by $$v_r=0$$ It
is seen that $V_θ$ is
infinite at the origin and decreases as Is this a contradiction? How is it that a vortex flow is irrotational? Note that the term "Irrotational" refers to the behaviour of a fluid element and not to the path taken by it. At an elemental level the flow is still irrotational. Such a vortex is called a Free Vortex. A familiar example is that of a bath tub vortex. In contrast to this is Solid Body rotation. This has a velocity given by $v_θ=Kr$, with a zero velocity at the origin. The velocity increases as one moves away from the origin. A cup of tea that is being continually stirred is a good example. It can be shown that the velocity potential for a solid body rotation is not uniquely defined and that such a forced flow is not a valid potential flow. A solid body rotation is not an irrotational flow. ## Circulation around a VortexThe circulation due to a vortex can be calculated as follows, which is non-zero for our irrotational flow. As there is a singularity in our region, namely at, It is usual to write the equation for velocity potential and stream
function in terms of circulation Note that for a positive value of circulation, the vortex flow is induced in an anti-clockwise direction. ## A Source-Sink PairConsidering a source and a sink placed at By taking tangents of the two sides (after manipulation), From the geometry it follows that Upon substituting, so that When
the distance between the source and the sink becomes smaller, i.e.,
when ( ## DoubletA Doublet is formed when the source and sink approach each other, i.e. $a→0$ and at the same time $m→∞$ such that ${ma} /π$ is constant, so that As a consequence the stream function becomes The velocity potential is Potential lines for a doublet are sketched above. It is seen that the streamlines are circles which are tangential to the x-axis while the equipotential lines are also circles but tangential to y-axis. ## Superposition of Elementary FlowsSimple flows such as uniform flow, source flow, vortex and doublet flow serve as primary components of potential flow. By combining these flows it is possiblt to build up more complicated flows. All flows discussed are linear. As the governing continuity equation is linear then if $A$ and $B$ are two solutions, then $mA±nB$ (where $m$ and $n$ are scale factors) is also a solution. The following programs canbe used to create streamlines and velocity potential lines for various cases of potential flow element superposition. __streamf.exe__-- Potential Flow stream line plotting program.*(MS windows executable)*__velpot.exe__-- Potential Flow velocity potential plotting program.*(MS windows executable)*__Potential Flow Arbitrary Stream Function plotting program.__
## Uniform Flow and a SourceIf a source is placed in the path of a uniform flow then the following flow is produced. The stream function and the velocity potential for the resulting flow are given by adding the two stream functions and velocity potentials as follows, $$ψ=V_∞y+m/{2π}θ$$ $$ψ=V_∞r\sin(θ)+m/{2π}θ$$ $$φ=φ(\text"uniform flow")+φ(\text"source flow")$$ $$φ=V_∞x+m/{2π}\ln(r)$$ $$φ=V_∞r\cos(θ)+m/{2π}\ln(r)$$ One of the interesting features to determine for the resulting flow is the creation of a stagnation point, i.e., point where the velocity goes to zero. It is clear that for this flow the stagnation point will occur on the x-axis. The location can be arrived at purely intuitively. The source produces a radial flow of magnitude while
the uniform flow produces a velocity of $V_∞$ in
the positive leading to At The equation for the streamline passing through the stagnation point is obtained as follows, thus The streamlines for this flow are shown in the above figure. The red stream line is the boundary between the two flow components and as the properties of a streamline are such that there is no flow across the line, then the red streamline could be treated as the boundary of a solid surface. In the present example, if we ignore the streamlines inside the "body" we have described the flow about a solid body. This body is referred to as a Rankine Half Body as it is "open" at the right hand end. Limits
of The velocity components for this flow are given by If the pressure in the free stream is $P_∞$, it follows from Bernoulli Equation that along a streamline around the body, thus allowing pressure to be calculated on the surface of the object. Usually in aerodynamic applications involving significant velocities and pressures any contribution due to elevation changes is negligible. The equation for pressure assumes a simple form, It is also possible to calculate the maximum velocity over the surface of the body. This occurs at the location $θ=63^{o}$ and is approximately equal to $1.26V_∞$. ## Rankine OvalIn the previous example, a half body open at one end was constructed. A closed body can be created by adding a trailing sink which will capture the mass flow from the source and thus close the rear of the shape. The stream function for this source/sink pair in a stream is given by or in cartesian coordinates, When the streamlines for this flow are plotted (see below) one
discovers that the The distance to the stagnation points from the origin or the Half Body Length is given by The other feature of interest, Half Width is found by determining the point of intersection of The solution for this non-linear equation is to be obtained by iteration. Rankine ovals include a wide range of bodies which can be obtained by varying the value of the parameter ${πV_∞a}/m$. 1These could be bodies stretched in any of the two directions. When stretched in x-direction one obtains elliptic bodies with a small half width compared to the span. The solution obtained could be a good approximation to the flow especially if viscous effects are small. On the other hand a considerable half width would indicate a bluff body prone to effects like separation. The solution obtained can hardly be accepted in this case. ## Flow Around a Circular CylinderFlow around a circular cylinder can be produced from the previous example by bringing the source and the sink closer together. In the limit this produces a uniform flow in combination with a doublet. The stream function and the velocity potential for this flow are given by, $$ψ=V_∞r\sin(θ)-{K\sin(θ)}/r\text" , "φ=V_∞r\cos(θ)+{K\cos(θ)}/r$$ The velocity components are given by, $$v_θ=-{∂ψ}/{∂r}=-\sin(θ)(V_∞+ K/r^2)$$ The radial velocity is zero when, This condition will identify the surface of the cylinder. Thus the radius
of the cylinder ( The equations for the streamline, velocity potential and the velocity
components can be re-written in terms of the size (radius = $$φ=V_∞r\cos(θ)(1+a^2/r^2)=V_∞x(1+a^2/r^2)$$ $$v_r=1/r{∂ψ}/{∂θ}=V_∞\cos(θ)(1-a^2/r^2)$$ $$v_θ=1/r{∂φ}/{∂θ}=-V_∞\sin(θ)(1+a^2/r^2)$$ The velocity components on the surface of the cylinder are obtained by
putting The stagnation point on the cylinder will this be at θ = 90 and ^{0}270. Note: the sign of the $v_θ$ component
is negative because of the definition of surface directions and polar
axes. The actual surface velocity at these maximum points will have a
magnitude of $2V_∞$ and
be in the same direction as the free stream flow.^{0}The surface pressure distribution is calculated from Bernoulli equation, Substituting for surface velocity gives, We can also express pressure in terms of pressure coefficient, C which gives, A symmetry about y-axis is apparent. When compared to the experimentally observed 90. The deviation from there to the
rear of the cylinder is because this region is dominated viscous
forces giving rise to separation. The pressure tends to remain
constant in the separated region and this has an effect upstream. For
High Re, a turbulent attached layer is present up to approximately
130^{0.}^{o}, but for low Re number laminar boundary separation occurs at 90^{o}
and the pressure field is different from the ideal.Symmetry in the theoretical Symmetric pressure loading for a non-rotating cylinder. ## Flow about a Lifting CylinderA lifting flow can be generated by adding a vortex to the flow about a cylinder just described. The assumed positive direction for rotation is clockwise so take care when looking at the sign of induced velocity components. The stream function and the velocity potential now become, $$φ=V_∞r\cos(θ)(1+a^2/r^2)-Γ/{2π}θ$$ Consequently the velocity components will be, At The stagnation points for this flow will be found by calculating locations where surface velocity is zero. $$\sin(β)=Γ/{4πV_∞a}$$ The surface velocity can now be written as ## Stagnation Points for a lifting circular cylinderThe stagnation points shown above are for the case when there is circulation
imposed on the cylinder is small was such that $Γ<4πV_∞a$. However
for larger values of circulation the angle ## Surface Pressure Distribution and LiftThe surface pressure is calculated from the Bernoulli equation as or The C The
magnitude of the lift force, $$L=∫_0^{2π}(-P\sin(θ)).a.dθ$$ $$L=∫_0^{2π}(-P_∞-1/2ρV_∞^2+2ρV_∞^2(\sin(θ)+\sin(β))^2)\sin(θ).a.dθ$$ The components from stream static pressure and velocity will be balanced around the cylinder and their integration will be zero. $$L=∫_0^{2π}(2ρV_∞^2(\sin^2(θ)+2\sin(θ)\sin(β)+\sin(β)^2)\sin(θ).a.dθ$$ Integration of the $\sin^2(β)\sin(θ);$ term and the $\sin^3(θ)$ from $0→2π$ also results in zero. $$L=4ρV_∞^2a\sin(β)π$$ when substituing for the expression relating $\sin(β)$ and $Γ$ ## Magnus EffectA force is produced when circulation is imposed upon
a cylinder placed in uniform flow. This force is the lift. This
effect is called Magnus Effect in honour of the scholar ## Kutta-Joukowsky TheoremThe result derived above, namely, $\text"Lift"=ρV_∞Γ$ is a very general one and is valid for any closed body placed in a
uniform stream that is creating circulation. It is named the Kutta-Joukowsky theorem in honour of
The theorem finds considerable application in calculating lift around aerofoils and will be used extensively in later sections covering aerofoil and wing analysis. |