NPTEL – Mechanical – Principle of Fluid Dynamics
Module 3 : Lecture 1 INVISCID INCOMPRESSIBLE FLOW (Fundamental Aspects)
In general, fluids have a well-known tendency to move or flow. The slight change in shear stress or appropriate imbalance in normal stresses will cause fluid motion. Fluid kinematics deals with various aspects of fluid motion without concerning the actual force that causes the fluid motion. In this particular section, we shall consider the ‘field’ concept to define velocity/ acceleration of fluid by virtue of its motion. In the later part, some ‘visualization’ concepts are introduced to define the motion of the fluid qualitatively as well as quantitatively.
There are two general approaches in analyzing the fluid motion. In the first method (Lagrangian approach), the individual fluid particles are considered and their properties are studied as a function of time. In the second method (Eulerian approach), the ‘field’ concept is introduced and the properties are completely prescribed as the functions of space and time. In other words, the attention is focused at fixed points in space as the fluid passes those points.
Velocity and Acceleration Field
Since the ‘continuum’ assumption holds well for fluids, the description of any fluid property (such as density, pressure, velocity, acceleration etc.) can be expressed as a function of location. These representation as a function of spatial coordinates is called as “field representation” of the flow. One of the most important fluid variables is the velocity field. It is a vector function of position and time with components uv, and w as scalar variables i.e. V=++ u( xyzt,,,) iˆˆ v( xyzt ,,,) j w( xyzt ,,,) kˆ (3.1.1) The magnitude of the velocity vector i.e. V=( uvw22 ++ 2) , is the speed of fluid. The total time derivative of the velocity vector is the acceleration vector field (a) of
the flow which is given as,
dV d{ u( xyzt,,,)} d{ v( xyzt ,,,)} d{ w( xyzt,,,)} a==++ ijkˆˆ ˆ (3.1.2) dt dt dt dt
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For instance, the scalar time derivative of u is expressed as,
d{ u( xyzt,,,)} ∂∂u u dx ∂ u dy ∂ u dz =+++ dt∂∂ t t dt ∂ t dt ∂ t dt ∂∂∂uuu ∂ u =+++uvw (3.1.3) ∂∂∂ttt ∂ t ∂u = +∇(Vu. ) ∂t
When u is replaced with vwand in the above equation, then the corresponding
expressions would be,
d{ v( xyzt,,,)} ∂v =+∇(Vv. ) dt∂ t (3.1.4) d{ w( xyzt,,,)} ∂w = +∇(Vw. ) dt∂ t
Now, summing them into a vector quantity, one may write Eq. (3.1.2) in compact form as, dV∂ V ∂∂ V V ∂∂ V V a= = + u + v + w = +∇( VV. ) dt∂ t ∂∂ x y ∂∂ z t (3.1.5) ∂∂ ∂ ∂∂∂ where, Vu= + v + w and ∇=iˆˆ + j + kˆ ∂∂xy ∂ z ∂∂ xy ∂ z ∂V In the above equation, is called as “local acceleration” and the second part i.e. ∂t ∂∂VV ∂ V uvw++ is called a “convective acceleration”. The total time derivative ∂∂xy ∂ z is called as “substantial/material” derivative. This field concept can be applied to any variable (vector or scalar). For example, one may write the total derivative for pressure and temperature field as,
dp∂ p ∂∂ p p ∂∂ p p = +u + v + w = +∇( Vp. ) dt∂ t ∂∂ x y ∂∂ z t (3.1.6) dT∂ T ∂∂ T T ∂∂ T T = +u + v + w = +∇( VT. ) dt∂ t ∂∂ x y ∂∂ z t
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Visualization of Fluid Flow
The quantitative and qualitative information of fluid flow can be obtained through sketches, photographs, graphical representation and mathematical analysis. However, the visual representation of flow fields is very important in modeling the flow phenomena. In general, there are four basic types of line patterns used to visualize the flow such as timeline, pathline, streakline and streamlines. Regardless of how the results are obtained, (i.e. analytically/experimentally/computationally) it is necessary to plot the data to get the feel of flow parameters that vary in time and/or shape (such as profile plots, vector plots and contour plots).
(a) Timeline: A ‘timeline’ is a set of fluid particles that form a line at a given instant (Fig. 3.1.1-a). Thus, it is marked at same instant of time. Subsequent observations of the line provide the information of the flow field. They are particularly useful in situations where uniformity of flow is to be examined.
(b) Pathline: It is the actual path traversed by a given fluid particle as it flows from one point to another (Fig. 3.1.1-b). Thus, the pathline is a Lagrangian concept that can be produced in the laboratory by marking the fluid particle and taking time exposure photograph of its motion. Pathlines can be calculated numerically for a known velocity field (V ) i.e.
tend = + x xstart ∫ V dt (3.1.7) tstart
(c) Streakline: A streakline consists of all particles in a flow that has previously passed through a common point (Fig. 3.1.1-c). Here, the attention is focused to a fixed point in space (i.e. Eulerian approach) and identifying all fluid particles passing through that point. These lines are laboratory tool rather than analytical tool. They are obtained by taking instantaneous photographs of selected particles that have passed through a given location in the flow field.
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(d) Streamline: These are the lines drawn in the flow field so that at a given instant, they are tangent to the direction of flow at every point in the flow field (Fig. 3.1.1-d). Since the streamlines are tangent to the velocity vector at every point in the flow field, there can be no flow across a streamline. Mathematically, these lines are obtained analytically by integrating the equations defining lines tangent to the velocity field. In a two dimensional flow field as shown in the figure, the slope of the streamline is equal to the tangent of the angle that velocity vector makes with x-axis i.e.
dy v = (3.1.8) dx u
This equation can be integrated to obtain the equation of streamlines.
When bundles of streamlines are considered in a flow field, it constitutes a ‘stream tube’ (Fig. 3.1.1-e). Since streamlines are everywhere parallel to the local velocity, fluid cannot cross a streamline, so fluids within a stream tube remain there and cannot cross the boundary at stream tube.
Fig. 3.1.1: Basic line patterns in fluid flow: (a) Timelines; (b) Pathline; (c) Streakline; (d) Streamline; (e) Streamtube.
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The following observations can be made about the fundamental line patterns;
1. Mathematically, it is convenient to calculate a streamline while other three are easier to generate experimentally. 2. The streamlines and timelines are instantaneous lines while pathlines and streakline are generated by passage of time. 3. In a steady flow, all the four basic line patterns are identical. Since, the velocity at each point in the flow field remains constant with time, consequently streamline shapes do not vary. It implies that the particle located on a given streamline will always move along the same streamline. Further, the consecutive particles passing through a fixed point in space will be on the same streamline. Hence, all the lines are identical in a steady flow. They do not coincide for unsteady flows. (f) Graphical data analysis techniques: Profile plots, vector plots and contour plots are few important techniques in which fluid flow properties can be analyzed. The profile plot (Fig. 3.1.2-a) indicates the variation of any scalar property (such as pressure, temperature and density) along some desired direction in a flow field. Using this plot, it is possible to examine the relative behavior of all variables in a multivariate data set.
Fig. 3.1.2: Graphical representation of data analysis technique: (a) profile plot, (b) vector plot, (c) contour plot.
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A vector plot (Fig. 3.1.2-b) is an array of arrows indicating the magnitude and direction of a vector property at an instant of time. Although, streamlines indicate the direction of instantaneous velocity field, but does not directly indicate the magnitude of velocity. A useful flow pattern for both experimental and computational fluid flow is the ‘vector plot’ that indicates the magnitude and direction of instantaneous vector property.
A contour plot (Fig. 3.1.2-c) is a two-dimensional plot of a three-dimensional surface showing lines where the surface intersects planes of constant elevation. Thus, they are curves with constant values of scalar property (or magnitude of vector property) at an instant of time. They can be filled in with either colors or sheds of gray representing the magnitude of the property.
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Module 3 : Lecture 2 INVISCID INCOMPRESSIBLE FLOW (Kinematic Description of Fluid Flow)
In order to illustrate this concept, we consider a typical fluid element of certain volume at any arbitrary time as shown in Fig. 3.2.1. After certain time interval, it has moved and changed its shape as well as orientation drastically. However, when we limit our attention to an infinitesimal particle of volume dV(= δδδ x.. y z) at time t
and (tt+δ ) within the fluid element, it may be observed that the change of its shape is limited to only stretching/shrinking and rotation with its sides remaining straight even though there is a drastic change in the finite fluid element. Thus, the particle motion in a fluid flow can be decomposed into four fundamental components i.e. translation, rotation, linear strain and shear strain as shown in Fig. 3.2.2. When the fluid particle moves in space from one point to another, it is referred as translation. Rotation of the fluid particle can occur in any of the orthogonal axis. In the case of linear strain, the particle’s side can stretch or shrink. When the angle between the sides of the particle changes, it is called as shear strain.
Fig. 3.2.1: Schematic representation of motion of finite fluid element and infinitesimal particle mass at two different time
steps.
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Fig. 3.2.2: Basic deformations of fluid mass: (a) Linear deformation; (b) Angular deformation.
Linear Motion and Deformation
Translation is the simplest type of fluid motion in which all the points in the fluid element have same velocity. As shown in Fig. 3.2.3-a, the particle located as point O will move to O’ during a small time interval δt . When there is a presence of velocity gradient, the element will tend to deform as it moves. Now, consider the effect of single velocity gradient (∂∂ux) on a small cube having sidesδδxy, and δ z and
volume δV= δδδ xyz. As shown in Fig. 3.2.3-b, the x-component of velocity of O and B is u . Then, x-component of velocity of points A and C would be,
u+∂( ux ∂)δ x, which causes stretching of AA’ by an amount (∂∂u x)δδ xt as shown in Fig.3.2.3-c. So, there is a change in the volume element ∂u δV= δ x( δδ yz) δ t ∂x
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Fig. 3.2.3: Linear deformation of a fluid element.
Rate at which the volume δ V changes per unit volume due to the velocity gradient (∂∂ux) is
1 d(δδ V) (∂∂ ux) t ∂u = lim = (3.2.1) δδV dt δt→0 t∂ x
In the presence of other velocity gradients (∂∂vy)and ( ∂ wz ∂) , Eq. (3.2.1) becomes,
1 dV(δ ) ∂∂∂uvw =++ =∇.V (3.2.2) δ V dt∂∂∂ x y z
If we look closely to the unit of velocity gradients (∂∂ux),( ∂∂ vy) and ( ∂ wz ∂) , then
they resemble to unit of strain rate and the deformation is associated in the respective directions of orthogonal coordinates in which the components of the velocity lie. Thus, the linear strain (Fig. 3.2.2-a) is defined as the rate of increase in length to original length and the linear strain rates are expressed as,
∂uvw ∂∂ εεε=,, = = (3.2.3) xx ∂∂∂xyzyy zz
The volumetric strain rate/volumetric dilatation rate is defined as the rate of increase of volume of a fluid element per unit volume.
1 dV ∂∂∂u v w =++=++εεε (3.2.4) V dt xx yy zz ∂∂∂x y z
In an incompressible fluid, the volumetric dilatation rate is zero because the fluid element volume cannot change without change in fluid density.
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Angular Motion and Deformation
The variations of velocity in the direction of velocity is represented by the partial derivatives (∂∂ux),( ∂∂ vy) and ( ∂ wz ∂) , which causes linear deformation in the
sense that shape of the fluid element does not change. However, cross variations of derivatives such as (∂∂uy),,( ∂∂ vz) ( ∂ wx ∂) will cause the fluid element to rotate.
These motions lead to angular deformation which generally changes the shape of the element.
Fig. 3.2.4: Angular deformation of a fluid element.
Let us consider the angular motion in x-y plane in which the initial shape is given by OACB, as shown in Fig. 3.2.4-a. The velocity variations cause the rotation and angular deformation so that the new positions become OA’ and OB’ after a time interval δt . Then the angles AOA’ and BOB’ are given by δαand δβ , respectively as shown in Fig. 3.2.4-b. Thus, the angular velocities of line OA and OB are,
δα ∂∂vuδβ ωωOA =lim = ;OB = lim = δδtt→→00δδtx∂∂ ty (∂∂v x)δδ xt ∂v For smallangles, tanδα≈= δα =δt (3.2.5) δ xx∂ (∂∂u y)δδ yt ∂u and tanδβ≈= δβ =δt δ yy∂
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When, both (∂∂vx) and ( ∂∂ uy) are positive, then both ωωOAand OB will be in
counterclockwise direction. Now, the rotation of the fluid element about z-direction
(i.e x-y plane) ωz can be defined as the average of ωωOAand OB . If counterclockwise rotation is considered as positive, then
1 ∂∂vu ωz = − (3.2.6) 2 ∂∂xy
In a similar manner, the rotation of the fluid element about x and y axes are denoted as
ωωxyand , respectively.
11∂∂wv ∂∂ uw ωωxy=−=−; (3.2.7) 22∂∂yz ∂∂ zx These three components can be combined to define the rotation vector (ω) in the form
as,
ˆˆˆ ωω=++xyi ω jk ω z 11∂wv ∂ ∂∂ uw 1 ∂∂ vu or, ω = −+iˆˆ − jk + −ˆ (3.2.8) 22∂∂yz ∂∂ z x 2 ∂∂ xy iˆˆ jkˆ 11∂∂ ∂ or, ω = = ∇×V 22∂∂xy ∂ z uv w
It is observed from Eq.(3.2.6) that the fluid element will rotate about z-axis, as an
undeformed block, only when, ωωOA=− OB i.e. ( ∂vx ∂) =−∂( uy ∂) . Otherwise it will be associated with angular deformation which is characterized by shear strain rate. When the fluid element undergoes shear deformation (Fig. 3.2.2-b), the average shear strain rates expressed in different cartesian planes as,
111∂∂vu ∂∂ wv ∂∂ wu εεεxy =+=+=+;;yz zx (3.2.9) 222∂∂xy ∂∂ yz ∂∂ z x
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Strain rate as a whole constitute a symmetric second order tensor i.e.
εεε xx xy xz εij= εεε yx yy yz (3.2.10) εεε zx zy zz
Vorticity
In a flow field, vorticity is related to fluid particle velocity which is defined as twice of rotation vector i.e.
∂wv ∂ ∂∂ uw ∂∂ vu ζω=2 =∇×V = − iˆˆ + − jk + − ˆ (3.2.11) ∂∂yz ∂∂ z x ∂∂ xy
Thus, the curl of the velocity vector is equal to the vorticity. It leads to two important definitions: . If ∇×V ≠0 at every point in the flow, the flow is called as rotational. It implies that the fluid elements have a finite angular velocity. . If ∇×V =0at every point in the flow, the flow is called as irrotational. It implies that the fluid elements have no angular velocity rather the motion is purely translational.
Irrotational Flow In Eq.(3.2.11) , if ∇×V =0 is zero, then the rotation and vorticity are zero. The flow fields for which the above condition is applicable is known as irrotational flow. The condition of irrotationality imposes specific relationship among the velocity gradients which is applicable for inviscid flow. If the rotations about the respective orthogonal axes are to be zero, then, one can write Eq. (3.2.11) as,
∂∂vu ∂∂ uw ∂ wv ∂ ωω=⇒=0 ;0 =⇒= ;0 ω =⇒= (3.2.12) zy∂∂xy ∂∂ z x x ∂∂ yz
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A general flow field would never satisfy all the above conditions. However, a uniform flow field defined in a fashion, for which uU= (a constant); v= 0; w = 0 , is certainly an example of an irrotational flow because there are no velocity gradients. A fluid flow which is initially irrotational may become rotational if viscous effects caused by solid boundaries, entropy gradients and density gradients become significant.
Circulation
It is defined as the line integral of the tangential velocity component about any closed curve fixed in the flow i.e.
Γ=∫ V. ds (3.2.13) where, ds is an elemental vector tangent to the curve and with length ds with counterclockwise path of integration considered as positive. For the closed curve path OACB as shown in Fig. 3.2.4-a, we can develop the relationship between circulation and vorticity as follows;
∂∂vu δδδδδδδΓ=ux + v + x y − u + y xvy − ∂∂xy ∂∂vu or, δΓ= −δδxy =2 ωδδz xy (3.2.14) ∂∂xy Then,Γ=V . ds = 2ωz dA = ∇× V dA ∫∫ ∫( )z AA
Hence, circulation around a closed contour is equal to total vorticity enclosed within it. It is known as Stokes theorem in two dimensions.
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Module 3 : Lecture 3 INVISCID INCOMPRESSIBLE FLOW (Stream Function and Velocity Potential)
Basic Equations of Fluid Motion
The differential relations for fluid particle can be written for conservation of mass, momentum and energy. In addition, there are two state relations for thermodynamic properties. They can be summarized as;
∂ρ Continuity:+∇ .(ρV ) = 0 ∂t dV Momentum: ρρ=gp −∇ +∇. τ dt ij (3.3.1) ∂e Energy: ρ +p( ∇.. V) =∇( kT ∇) +Φ ∂t Thermodynamicstate relations:ρρ=( pT ,) ; e = e( pT , )
Here, Φ is the viscous-dissipation function, e is the internal energy and k is the thermal conductivity of the fluid. In general, the density is a variable and all these equations have 5-unknown parameters i.e. ρ,Vpe , , and T. In an incompressible flow with constant viscosity, the momentum equation can be decoupled from energy equation. Thus, continuity and momentum equations are solved simultaneously for pressure and velocity. However, there are certain flow situations, which can wipe out continuity equation by defining a suitable variable (called as stream function) and thereby solving the momentum equation with a single variable.
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Stream Function
The idea of introducing stream function works only if the continuity equation is reduced to two terms. There are 4-terms in the continuity equation that one can get by expanding the vector equation (3.3.1) i.e.
∂ρ ∂∂∂(ρρρuvw) ( ) ( ) +++ =0 (3.3.2) ∂∂tx ∂ y ∂ z
For a steady, incompressible, plane, two-dimensional flow, this equation reduces to,
∂∂uv +=0 (3.3.3) ∂∂xy
Here, the striking idea of stream function works that will eliminate two velocity components uvand into a single variable (Fig. 3.3.1-a). So, the stream function {ψ ( xy, )} relates to the velocity components in such a way that continuity equation (3.3.3) is satisfied.
∂∂ψψ uv=; = − ∂∂yx (3.3.4) ∂∂ψψ or,Vij=ˆˆ − ∂∂yx
Fig. 3.3.1: Velocity components along a streamline.
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Fig. 3.3.2: Flow between two streamlines.
The following important points can be noted for stream functions;
1. The lines along which ψ is constant are called as streamlines. In a flow field, the tangent drawn at every point along a streamline shows the direction of velocity (Fig. 3.3.1-b). So, the slope at any point along a streamline is given by,
dy v = (3.3.5) dx u
Referring to the Fig. 3.3.2-a, if we move from one point ( xy, ) to a nearby point ( x++ dx, y dy) , then the corresponding change in the value of stream function is dψ which is given by,
∂∂ψψ dψ = dx + dy =−+ v dx u dy (3.3.6) ∂∂xy
Along a line of constant ψ ,
dψ =−+= v dx u dy 0 dy v (3.3.7) or, = dx u
The Eq. (3.3.5) is same as that of Eq. (3.3.7). Hence, it is the defining equation for the streamline. Thus, infinite number streamlines can be drawn with constant ψ . This family of streamlines will be useful in visualizing the flow patterns. It may also be noted that streamlines are always parallel to each other.
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2. The numerical constant associated toψ , represents the volume rate of flow.
Consider two closely spaced streamlines ψand ( ψψ+ d ) as shown in Fig. 3.3.2-a. Let dq represents the volume rate of flow per unit width perpendicular to x-y plane, passing between the streamlines. At any arbitrary surface AC, this volume flow must be equal to net outflow through surfaces AB and BC. Thus,
∂∂ψψ dq =−+= v dx u dy dx + dy = dψ ∂∂xy (3.3.8) or, dq = dψ
Hence, the volume flow rate (q) can be determined by integrating Eq. (3.3.8) between
ψψ streamlines 12and as follows;
ψ 2 =ψψ = − ψ qd∫ 21 (3.3.9) ψ1
So, the change in the value of stream function is equal to volume rate of flow. If the upper streamline ψ 2 has a value greater than the lower one ψ1 , then the volume flow rate is positive i.e. flow takes place from left to right (Fig. 3.3.2-b).
3. In cylindrical coordinates, the continuity equation for a steady, incompressible, plane, two-dimensional flow, reduces to
11∂ (rv ) ∂ v r +=θ 0 (3.3.10) rrr∂∂θ
The respective velocity components vvr and θ are shown in Fig. 3.3.1-c. The stream function {ψθ(r, )}that satisfies Eq. (3.3.10), can then be defined as,
1 ∂∂ψψ vv=; = − (3.3.11) r rr∂∂θ θ
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4. In a steady, plane compressible flow, the stream function can be defined by including the density of the fluid. But, the change in the stream function is equal to mass flow rate(m ) .
∂∂ψψ ρρuv=; = − ∂∂yx ∂∂ψψ dm =−+ρρ v dx u dy = dx + dy = dψ (3.3.12) ∂∂xy or, dm = dψ
5. One important application in a two-dimensional plane is the inviscid and irroational flow where, there is no velocity gradient and ωz = 0 . Then, the vorticity vector becomes,
ˆˆ∂∂vu ζω==−=20z kk ∂∂xy ∂ ∂ψψ ∂∂ or, −−kˆ =0 ∂x ∂ x ∂∂ yy (3.3.13) ∂∂22ψψ or, −=0 ∂∂xy22 or,∇=2ψ 0
This is a second order equation and is quite popular in mathematics and is known as Laplace equation in a two-dimensional plane.
Velocity Potential
An irrotational flow is defined as the flow where the vorticity is zero at every point. It gives rise to a scalar function (φ ) which is similar and complementary to the stream
function(ψ ) . Let us consider the equations of irrortional flow and scalar function (φ ) . In an irrotational flow, there is no vorticity (ξ )
ξ =∇×V =0 (3.3.14)
Now, take the vector identity of the scalar function (φ ) ,
∇×( ∇φ ) =0 (3.3.15)
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i.e. a vector with zero curl must be the gradient of a scalar function or, curl of the gradient of a scalar function is identically zero. Comparing, Eqs. (3.3.14) and (3.3.15), we see that, V = ∇φ (3.3.16)
Here, φ is called as velocity potential function and its gradient gives rise to velocity vector. The knowledge φ immediately gives the velocity components. In cartesian coordinates, the velocity potential function can be defined as, φφ= ( xyz,,) so that
Eq. (3.3.16) can be written as,
∂∂∂φφφ uiˆˆ++ v j wkˆˆ =i ˆ + ˆj + k (3.3.17) ∂∂xyz ∂
So, the velocity components can be written as,
∂∂∂φφφ uvw=;; = = (3.3.18) ∂∂∂xyz
In cylindrical coordinates, if φφ= (rz,, θ) , then
∂φ ∂∂ φφ VVV=;; = = (3.3.19) rz∂∂∂rzθ θ
Further, if the flow is incompressible i.e. ρρ=constant and( ∂ ∂=t) 0, then
continuity equation can be written as,
∂ρ +∇.0(ρV ) = ∂ t or,ρ (∇= .V ) 0 (3.3.20) or,∇= .V 0
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Therefore, for a flow which is incompressible and irrotational, Eqs. (3.3.16) and (3.3.20) can be combined to yield a second order Laplace equation in a three- dimensional plane.
∇∇.0( φ ) = or,∇=2φ 0 (3.3.21) ∂∂∂222φφφ or, ++=0 ∂∂∂xyz2 22
Thus, any irrotational, incompressible flow has a velocity potential and stream function (for two-dimensional flow) that both satisfy Laplace equation. Conversely, any solution of Laplace equation represents both velocity potential and stream function (two-dimensional) for an irrotational, incompressible flow.
An irrotational flow allows a velocity potential to be defined and leads to simplification of fundamental equations. Instead of dealing with the velocity components uv, and was unknowns, one can deal with only one parameterφ , for a given problem. Since, the irrotational flows are best described by velocity potential, such flows are called as potential flows. In these flows, the lines with constant φ , is known as equipotential lines. In addition, a line drawn in space such that ∇φ is the tangent at every point is defined as a gradient line and thus can be called as streamline.
Stream Function vs Velocity Potential
The velocity potential is analogous to stream function in a sense that the derivatives of both φψand yield the flow field velocities. However, there are distinct differences between φψand :
. The flow field velocities are obtained by differentiating φ in the same direction as the velocities, whereas, ψ is differentiated normal to the velocity direction. . The velocity potential is defined for irrotational flows only. In contrast, stream function can be used in either rotational or irrotational flows. . The velocity potential applies to three-dimensional flows, whereas the stream function is defined for two dimensional flows only.
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It is seen that the stream lines are defined as lines of constant ψ which are same as gradient lines and perpendicular to lines of constant φ . So, the equipotential lines and stream lines are mutually perpendicular. In order to illustrate the results more clearly, let us consider a two-dimensional, irrotational, incompressible flow in Cartesian coordinates.
For a streamline, ψ ( xy,) = constant , and the differential of ψ is zero.
∂∂ψψ dψ =+= dx dy 0 ∂∂xy or, dψ =−+= v dx u dy 0 (3.3.22) dy v or, = dxψ =constant u
Similarly, for an equipotential line, φ ( xy,) = constant , and the differential of φ is
zero.
∂∂φφ dφ =+= dx dy 0 ∂∂xy or, dφ =+= u dx v dy 0 (3.3.23) dy u or, = − dx φ =constant v
Combining Eqs. (3.3.22) and (3.3.23), we can write,
dy 1 = − (3.3.24) dx ψ =constant (dy dx)φ =constant
Hence, the streamlines and equipotential lines are mutually perpendicular.
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Module 3 : Lecture 4 INVISCID INCOMPRESSIBLE FLOW (Basic Potential Flows - I)
Potential Theory
In a plane irrotaional flow, one can use either velocity potential or stream function to define the flow field and both must satisfy Laplace equation. Moreover, the analysis of this equation is much easier than direct approach of fully viscous Navier-Stokes equations. Since the Laplace equation is linear, various solutions can be added to obtain other solutions. Thus, if we have certain basic solutions, then they can be combined to obtain complicated and interesting solutions. The analysis of such flow field solutions of Laplace equation is termed as potential theory. The potential theory has a lot of practical implications defining complicated flows. Here, we shall discuss the stream function and velocity potential for few elementary flow fields such as uniform flow, source/sink flow and vortex flow. Subsequently, they can be superimposed to obtain complicated flow fields of practical relevance.
Governing equations for irrotational and incompressible flow
The analysis of potential flow is dealt with combination of potential lines and streamlines. In a planner flow, the velocities of the flow field can be defined in terms of stream functions {ψ ( xy, )}and potential functions{φ ( xy, )} as,
∂∂{ψψ( xy,,)} { ( xy)} uv= ; = − ∂∂yx (3.4.1) ∂∂{φφ( xy,,)} { ( xy)} uv= ; = ∂∂xy
The stream function ψ is defined such that continuity equation is satisfied whereas, for low speed irrotational flows(∇×V =0) , if the viscous effects are neglected, the continuity equation(∇=• V 0) , reduces to Laplace equation for φ . Both the functions
satisfy the Laplace equations i.e.
∂∂22φ φ ∂∂ 22 ψψ += += 220; 220 (3.4.2) ∂∂xy ∂∂ xy
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Thus, the following obvious and important conclusions can be drawn from Eq. (3.4.2);
• Any irrotational, incompressible and planner flow (two-dimensional) has a velocity potential and stream function and both the functions satisfy Laplace equation. • Conversely, any solution of Laplace equation represents the velocity potential or stream function for an irrotational, incompressible and two-dimensional flow. Note that Eq. (3.4.2) is a second-order linear partial differential equation. If there are
n separate solutions such as,φφ12( xy,) ,( xy ,) ,...... , φn ( xy , ) then the sum (Eq. 3.4.3) is also a solution of Eq. (3.4.2).
φφφ( xy,) =12( xy ,) +( xy ,) ++ ...... φn ( xy , ) (3.4.3)
It leads to an important conclusion that a complicated flow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementary flows which are also irrotational and incompressible. However, different values of φψor represent the different streamline patterns of the body and at the same time they satisfy the Laplace equation. In order to differentiate the streamline patterns of different bodies, it is necessary to impose suitable boundary conditions as shown in Fig. 3.4.1. The most common boundary conditions include far-field and wall boundary conditions. on the surface of the body (i.e. wall).
Fig. 3.4.1: Boundary conditions of a streamline body.
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Far away from the body, the flow approaches uniform free stream conditions in all directions. The velocity field is then specified in terms of stream function and potential function as,
∂∂ψφ uV= = = ∂∂yx∞ (3.4.4) ∂∂ψφ v =−==0 ∂∂xy
On the solid surface, there is no velocity normal to the body surface while the tangent at any point on the surface defines the surface velocity. So, the boundary conditions can be written in terms of stream and potential functions as,
∂∂φψ =0; = 0 (3.4.5) ∂∂ns
Here, s is the distance measured along the body surface and n is perpendicular to the body. Thus, any line of constant ψ in the flow may be interpreted as body shape for which there is no velocity normal to the surface. If the shape of the body is given by y= fx( ) , then ψψ= =constant is alternate boundary condition of Eq. (3.4.5). b body y= yb If we deal with wall boundary conditions in terms of uvand , then the equation of streamline evaluated at body surface is given by,
dy 1 v b =−= (3.4.6) dx ψ =constant (dyb dx)φ =constant u surface
It is seen that lines of constant φ (equi-potential lines) are orthogonal to lines of constant ψ (streamlines) at all points where they intersect. Thus, for a potential flow field, a flow-net consisting of family of streamlines and potential lines can be drawn, which are orthogonal to each other. Both the set of lines are laplacian and they are useful tools to visualize the flow field graphically.
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Referring to the above discussion, the general approach to the solution of irrotational, incompressible flows can be summarized as follows;
• Solve the Laplace equation for φψor along with proper boundary conditions. • Obtain the flow velocity from Eq. (3.4.1) • Obtain the pressure on the surface of the body using Bernoulli’s equation.
1122 p+=+ρρ Vp V∞ (3.4.7) 22∞
In the subsequent section, the above solution procedure will be applied to some basic elementary incompressible flows and later they will be superimposed to synthesize more complex flow problems.
Uniform Flow
The simplest type of elementary flow for which the streamlines are straight, parallel with constant velocity, is known as uniform flow. Consider a uniform flow in positive x-direction as shown in Fig. 3.4.2. This flow can be represented as,
uV=∞ ;0 v = (3.4.8)
Fig. 3.4.2: Schematic representation of a uniform flow.
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The uniform flow is a physically possible incompressible flow that satisfies continuity equation (∇=• V 0) and the flow is irrotational (∇×V =0) . Hence, the velocity
potential can be written as,
∂∂φφ =uV =;0 = v = ∂∂xy∞ (3.4.9) ∂∂ψψ =uV =;0 =−= v ∂∂yy∞
Integrating Eq.(3.4.9) with respect to x,
φφ=+=+Vx∞ f1( y); g 11( x) C
⇒=g1( x) Vx∞ and f11( y) = C ⇒=φ Vx + C ∞ 1 (3.4.10) ψψ=+=+ Vy∞ f2( x); g 22( y) C
⇒=g2( y) Vy∞ and f22( x) = C
⇒=ψ Vy∞ + C2
In practical flow problems, the actual values of φψand are not important, rather it is always used as differentiation to obtain the velocity vector. Hence, the constant appearing in Eq. (3.4.10) can be set to zero. Thus, for a uniform flow, the stream functions and potential function can be written as,
φψ=Vx∞∞; = Vy (3.4.11)
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Fig. 3.4.3: Flow nets drawn for uniform flow.
When Eq. (3.4.11), is substituted in Eq. (3.4.2), Laplace equation is satisfied. Further, if the uniform flow is at an angle θ with respect to x-axis as shown in Fig. 3.4.2, then the generalized form of stream function and potential function is represented as follows;
φ=+=−Vx∞∞( cos θ y sin θψ) ; Vy( cos θ x sin θ) (3.4.12)
The flow nets can be constructed by assuming different values of constants in Eq. (3.4.11) and with different angle θ as shown in Fig. 3.4.3. The circulation in a uniform flow along a closed curve is zero which gives the justification that the uniform flow is irrotational in nature.
Γ=• = = ∫∫V ds V∞ ds 0 (3.4.13) CC
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Source/Sink Flow
Consider a two-dimensional incompressible flow where the streamlines are radially outward from a central point ‘O’ (Fig. 3.4.4). The velocity of each streamlines varies inversely with the distance from point ‘O’. Such a flow is known as source flow and its opposite case is the sink flow, where the streamlines are directed towards origin. Both the source and sink flow are purely radial. Referring to the Fig. 3.4.4, if
vvr and θ are the components of velocities along radial and tangential direction respectively, then the equations of the streamlines that satisfy the continuity equation (∇=• V 0) are,
c vv=;0 = (3.4.14) r r θ
Here, the constant c can be related to the volume flow rate of the source. If we define Λ as the volume flow rate per unit length perpendicular to the plane, then,
Λ=(2π rv) r ΛΛ (3.4.15) or,vc= and = r 22ππr
The potential function can be obtained by writing the velocity field in terms of cylindrical coordinates. They may be written as,
∂φφ Λ∂1 =vv =;0 = = ∂∂rr 2πθ rr θ (3.4.16) 1 ∂ψψ Λ∂ ==vv;0 −== r∂∂θπr 2 ryθ
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Fig. 3.4.4: Schematic representation of a source and sink flow.
Integrating Eq. (3.4.16) with respect to r and θ , we can get the equation for potential function and stream function for a source and sink flow.
Λ φ=+=+lnr f( θφ) ; C gr( ) 2π 3 33 Λ ⇒=f(θ ) Cand gr( ) =ln r 3332π Λ ⇒=φ ln rC + 2π 3 (3.4.17) Λ ψθ=+=+fr( ); ψ C g( θ) 2π 4 44 Λ ⇒=fr( ) Cand g(θ ) = 4442π Λ ⇒=ψθ +C 2π 4
The constant appearing in Eqs (3.4.17) can be dropped to obtain the stream function and potential function.
ΛΛ φ= lnr ; ψθ= (3.4.18) 22ππ
This equation will also satisfy the Laplace equation in the polar coordinates. Also, it represents the streamlines to be straight and radially outward/inward depending on the source or sink flow while the potential lines are concentric circles shown as flow nets in Fig. 3.4.5. Both the streamlines and equi-potential lines are mutually perpendicular.
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It is to be noted from Eq. (3.4.15) that, the velocity becomes infinite at origin (r = 0) which is physically impossible. It represents a mathematical singularity where the continuity equation (∇=• V 0) is not satisfied. We can interpret this point as
discrete source/sink of given strength with a corresponding induced flow field about this point. Although the source and sink flows do not exist, but many real flows can be approximated at points, away from the origin, using the concept of source and sink flow.
Fig. 3.4.5: Flow nets drawn for of a source and sink flow.
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Module 3 : Lecture 5 INVISCID INCOMPRESSIBLE FLOW (Basic Potential Flows - II)
Doublet Flow
The third type of basic elementary flow is the combination of source-sink pair of equal strength in a special fashion. It leads to singularity called as doublet. Consider a source and sink pair of equal strength Λ and separated by a distance 2a as shown in Fig. 3.5.1. At a point ‘P’ in the flow field, the combined stream function of the pair can be written as,
Λ ψ=−−( θθ12) (3.5.1) 2π
Fig. 3.5.1: Combination of source and sink located along x-axis.
Eq. (3.5.1) can be re-written in the following form.
2πψ −=−(θθ12) Λ (3.5.2) 2πψ tanθθ12− tan ⇒tan − = tan (θθ12 −=) Λ+1 tanθθ12 tan
Referring to Fig. 3.5.1 and using the concepts of trigonometry, the following relations can be obtained i.e.
rrsinθθsin tanθθ= ; tan = (3.5.3) 12racosθθ−+ racos
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Substituting these results in Eq. (3.5.2), we can obtain,
2πψ 2ar sin θ tan −= Λ−ra22
Λ −1 2ar sinθ ⇒=−ψ tan (3.5.4) 2π ra22− Λ arsinθ ⇒=−ψ (For small values of a) π (ra22− )
Now, a doublet can be formed by bringing the source-sink pair as close to each other (a → 0) while increasing its strength (Λ→∞) and keeping the product (Λa π )
constant. Then, rr( 22−→ a) 1 rand Eq. (3.5.4) can be simplified as,
κθsin ψ = − (3.5.5) r
Here, κπ=( Λa 2 )is a constant and called as strength of the doublet. Since the stream function and potential function is mutually perpendicular, we can write the velocity potential for the doublet as,
κθcos φ = (3.5.6) r
Combining Eqs (3.5.5) and (3.5.6), the flow nets can be drawn as shown in Fig. 3.5.2.
Fig. 3.5.2: Flow nets drawn for doublet flow.
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The equation of streamline can be obtained from Eq.(3.5.5) for a doublet flow i.e.
κθsin ψ =−=constant (c) r (3.5.7) κ ⇒=r sinθ c
Fig. 3.5.3: Streamlines for doublet flow.
It may be seen from the Fig. (3.5.3) and Eq. (3.5.7) that the streamlines for a doublet κ flow are the family of circles with diameter if the source and sink are placed at c origin (a → 0) . The different circles correspond to the different values of the parameter c . The doublet has associated with a direction with which the flow moves around the circular streamlines. By convention, the direction of the doublet is shown by an arrow drawn from sink to the source so that Eq. (3.5.5) is consistent with respect to sign convention.
Vortex Flow
Three types of elementary flows (uniform flow, source/sink flow and doublet flow) have been discussed earlier. Now, the last elementary flow will be introduced called as vortex flow. Consider a flow field in which the streamlines are concentric circles about a given point which is exactly opposite case when the velocity potential and
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stream function for the source is interchanged. Here, the velocity along any given circular streamline is constant, while it can vary inversely with distance from one
streamline to another from a common center. Referring to the Fig. 3.5.4, if vvr and θ are the components of velocities along radial and tangential direction respectively, then the flow field can be described as given below,
c vv=0; = (3.5.8) r θ r
Fig. 3.5.4: Schematic representation of a vortex flow. It may be easily shown that streamlines satisfy the continuity equation i.e. ∇=• V 0 and the vortex flow is irrotational i.e. ∇×V =0 at every point except origin (r = 0) .
In order to evaluate the constant appearing in Eq. (3.5.8), let us take the circulation around a given streamline of radius r :
Γ=V ds =− vθ (2π r) ∫C Γ (3.5.9) ⇒=−v θ 2π r
It may be seen by comparing Eqs. (3.5.8) and (3.5.9) that
Γ cc=− ⇒Γ=−2π (3.5.10) 2π
Thus, the circulation taken about all the streamlines is the same value. So, it is called as the strength of the vortex flow while the velocity field is given by Eq. (3.5.9). It may be noted that vθ is negative when Γ is positive i.e. vortex of positive strength rotates in clockwise direction. Now, let us obtain the velocity potential and stream function for the vortex flow from the velocity field. By definition,
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∂φφ1 ∂Γ =vv =0; = = − ∂∂rrr θπθ 2 r (3.5.11) 1 ∂ψψ ∂Γ ==−==−vv0; r∂∂θπr rrθ 2
Integrating the above equations, the velocity potential and stream function are obtained as,
ΓΓ φ=−= θψ; ln r (3.5.12) 22ππ
Once again it is clear from this equation that streamlines (ψ = constant) for a vortex
flow is given by concentric circles with fixed radius while equipotential lines (φ = constant) are the straight radial lines from the origin with constant θ . Both streamlines and equipotential lines are mutually perpendicular as shown in Fig. 3.5.5.
Fig. 3.5.5: Flow nets drawn for of a free vortex flow.
The following general remarks may be made for the vortex flow;
• The vortex flow is irrotational everywhere in the flow field except at origin where the vorticity is infinite. Hence the origin r = 0 is a singular point in the flow field which may be interpreted as point vortex and it induces circular vortex flow as shown in the Fig. 3.5.4. The irrotational vortex is usually called as free vortex. The swirling motion of water as it drains out of a bathtub is an example of free vortex.
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• When the fluid particles rotate as a rigid body such that
vθ = cr11; c isa constant , then the vortex motion is rotational and it cannot be described with velocity potential. This type of rotational vortex is commonly called as forced vortex. The motion of a liquid contained in a tank, when rotated about its axis with angular velocity ω corresponds to a forced vortex. • A combined vortex is the one with a forced vortex as the central core and velocity distribution corresponding to that of free vortex outside the core. c v=>=≤2 ( r r); vω rr( r) (3.5.13) r r 00θ
Here, c2 and ω are constants and r0 is corresponds to the radius of the central core.
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Module 3 : Lecture 6 INVISCID INCOMPRESSIBLE FLOW (Superposition of Potential Flows - I)
Method of Superposition
The potential flows are governed by the linear partial differential equation commonly called as “Laplace Equation”. The elementary basic plane potential flows include uniform flow, source/sink flow, doublet flow and free vortex flow. The details of these flow fields have already been discussed and are summarized in the following Table 3.6.1. A variety of interesting potential flow can be obtained by combination of velocity potential and stream function of basic potential flows.
In an inviscid flow field, a streamline can be considered as a solid boundary because there is no flow through it. Moreover, the conditions along the sold boundary and the streamline are the same. Hence, the combinations of velocity potential and stream functions of elementary flows will lead to a particular body shape that can be interpreted as flow around that body. The method of solving such potential flew problems is commonly called as, method of superposition.
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Table 3.6.1: Summary of basic, plane potential flows
Description of Flow Field Velocity Potential Stream Function Velocity components
θ Uniform flow at an angle φ=Vx∞ ( cos θθ + y sin ) ψ=Vy∞ ( cos θθ − x sin ) uV= ∞ cosθ with x-axis vV= ∞ sinθ
Source Λ>0 or Sink Λ Λ Λ ( ) φ = ln r ψθ= v = 2π 2π r 2π r (Λ<0) vθ = 0
Doublet κθcos κθsin κθcos φ = ψ = − = − vr 2 r r r κθsin v = − θ r 2
Free Vortex Γ Γ v =0 φθ= − ψ = ln r r 2π 2π Γ Counter-clockwise (Γ<0) vθ = 2π r Clockwise(Γ>0)
Combination of a Uniform Flow with a Source
A source of strength Λ , located at origin is superimposed with a uniform stream with
velocity V∞ as shown in Fig. 3.6.1. The resulting stream function can be written as,
Λ ψψ=uniform += ψ source Vr∞ sin θ + θ (3.6.1) 2π
Fig. 3.6.1: Superposition of uniform flow and a source.
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The streamlines of the combined flow leads to the flow over a semi-infinite body and are obtained as,
Λ ψ=Vr∞ sin θθ += constant (3.6.2) 2π
The velocity field is obtained from stream function through differentiation i.e.
1 ∂ψψ Λ∂ vV= =cosθθ + ; v =−=− Vsin (3.6.3) r r∂∂θπ∞∞2 rrθ
From the Fig. 3.6.1, it is clear that the flow becomes stagnant at some point because the velocity due to the source will cancel with that of uniform flow. Hence, a stagnation point is created which is obtained by assigning the velocity components to zero value in Eq. (3.6.3).
Λ VVcosθθ+= 0; sin = 0 (3.6.4) ∞∞2π r
Solving for r and θ , the coordinates of stagnation point ‘B’ is found out i.e. Λ (rb,,θπ) = ( ) and the distance b = , directly upstream of the source. It may be 2πV∞ observed that the point ‘B’ will be blown further downstream, if source strength is
increased keeping V∞ same, and increasing V∞ , keeping source strength same. When the coordinates of ‘B’ is substituted in Eq.(3.6.1), the streamline ‘ABC’ passing through the stagnation point is obtained i.e.
Λ ΛΛ ψstagnation =V∞ sinππ +==constant (3.6.5) ππ 2V∞ 22
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Λ Since, πbV = , it follows that the equation of the streamline passing through the ∞ 2 stagnation point is obtained from Eq. (3.6.1) as follows;
Λ ψπ= = bV stagnation 2 ∞
πbV∞ ⇒V∞∞ r sinθ += θπbV (3.6.6) π b(πθ− ) ⇒=r ;θπ varies from 0 to 2 sinθ
The streamline ‘ABC’ following the equation (3.6.6) is shown in Fig. 3.6.1. The following important observations can be made from the figure;
• The streamline ‘ABC’ contains the stagnation point at ‘B’ and separates the flow coming from the free stream and fluid emanating from source. All the fluid outside ‘ABC’ are from the free stream while the fluid inside ‘ABC’ are from the source. Hence, the singularity in the flow field (i.e. source) occurs inside the body whereas there is no singularity in the free stream (outside ‘ABC’). • In inviscid flow, the velocity at the surface of the solid body is tangent to the body. So, any streamline of this combined flow field can be replaced by a solid surface of same shape. Hence, with respect to free stream, the flow would not feel the difference if the streamline ‘ABC’ is replaced with a solid Λ body. The streamline ψ = extends downstream to infinity, forming a stagnation 2 semi-infinite body and is called as ‘Rankine Half-Body”. • Referring to Eq. (3.6.6), it is seen that the width of the half body yb=(πθ − )
asymptotically approaches to 2πb while the half-width is given by ± πb when θπ→ 0 or 2 . • For the half-body shown in Fig. 3.6.1, the magnitude of velocity at any point is given by the following equation;
2 2 22 2V∞Λ cosθ ΛΛ V=+= vvVr θ ∞ + +and b= ππrr22 π V ∞ (3.6.7) 2bb2 ⇒=22 +θ + VV∞ 1 cos 2 rr
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Using the Bernoulli’s equation, the pressure any arbitrary point on the half-body
can be obtained with the knowledge of free stream pressure ( p∞ ) and velocity
(V∞ ) .
11 p+=+ρρ Vp22 V (3.6.8) ∞22 ∞∞
Here, the elevation change is neglected.
Combination of a Uniform Flow with a Source-Sink Pair
The superposition of a uniform flow with a source results in the flow over the semi- infinite body ‘ABC’ as shown in Fig. 3.6.1. This is a half-body that stretches to infinity in the downstream and the body is not closed. If a sink of equal strength is added to this flow in the downstream and equally spaced from origin as that of source, then the resulting shape will be closed. The resulting flow field is shown in Fig. 3.6.2.
Fig. 3.6.2: Superposition of uniform flow and a source-sink pair.
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Let us consider a source and sink with strengths Λand −Λequally spaced at a distance b from the origin. When a uniform stream of velocity V∞ is superimposed on it, the combined flow at any point in the flow can be written as follow;
Λ ψψ= + ψ += ψVrsin θ +( θθ −) uniform source sink ∞ 2π 1 2 Λ 2br sinθ ψθ= − −1 or,Vr∞ sin tan 22 (3.6.9) 2π rb−
Λ −1 2by or, ψ =Vy∞ − tan 2π xyb2+− 22
From the geometry of Fig. 3.6.2, it is seen that θθ12and are the functions of rb,θ and . Assigning the velocity components to zero value, two stagnation points can be obtained in similar manner and they are located at points ‘A and B’ in the Fig. 3.6.2. Their distances from origin can also be calculated.
Λb OA= OB = l = b2 + πV∞ (3.6.10) l Λ ⇒=1 + bπ Vb∞
The equation of the streamline is given by,
Λ ψ=Vr∞ sin θ +( θθ12 −=) constant (3.6.11) 2π
The equation of specific streamlines passing through the stagnation points
A (θθ=12 = θ = π) and B( θθ=12 = θ = 0) and B(θθ=12 = θ = 0)’ is obtained by assigning the constant appearing in Eq. (3.6.11) as 0.
Λ ψ=Vr∞ sin θ +( θθ12 −=) 0 (3.6.12) 2π
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The body-half width (h) can be obtained by determining the value y where the y-axis intersects streamline ψ = 0. , Thus, from Eq. (3.6.9), one can obtain the body half width with yh= .
2 hh12 πVb∞ h = −1 tan (3.6.13) bb2 Λ b
From the above mathematical analysis the following physical interpretation can be made;
• The stagnation streamline described by Eq. (3.6.12), is the equation of an oval and is the dividing streamline. This particular shape is called as “Rankine Oval”. All the flow from the source is consumed by the sink and is contained entirely inside the oval. The flow outside the oval is originated through uniform flow only and can be interpreted as inviscid, irrotational and incompressible flow over solid body. Also, the potential solution for the Rankine oval gives the reasonable approximation of velocity outside the thin, viscous boundary layer and pressure distribution on the front part of the body. • Using Eqs. (3.6.10) and (3.6.13), a large variety of body shapes with different length to width ratio can be obtained for different values of the parameter
Vb∞ . As this parameter becomes large, the flow around a slender body is Λ described while the smaller values give the flow field around a blunt shape body.
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Module 3 : Lecture 7 INVISCID INCOMPRESSIBLE FLOW (Superposition of Potential Flows - II)
Non-Lifting Flow over a Circular Cylinder
It is seen earlier that flow over a semi-infinite body can be simulated by combination of a uniform flow with a source and flow over an oval-shaped body can be constructed by superimposing a uniform flow and a source-sink pair. A circular cylinder is one of the basic geometrical shapes and the flow passing over it can be simulated by combination of a uniform flow and doublet. When the distance between source-sink pair approaches zero, the shape Rankine oval becomes more blunt and approaches a circular shape.
Consider the superposition of a uniform flow of velocity V∞ and a doublet of strength Λ as shown in Fig. 3.7.1. The direction of the doublet is upstream, facing into uniform flow.
Fig. 3.7.1: Superposition of a uniform flow and doublet.
The stream function for the combined flow is,
ΛΛsinθ ψψ=+= ψ θ − =θ − uniform doublet Vr∞∞sin Vrsin 1 2 22ππr Vr∞ (3.7.1) R2 Λ ⇒=ψθ − = Vr∞ sin 12 ; R rV2π ∞
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The velocity field is obtained as,
1 ∂∂ψψRR22 = =− θθ=−=−+ vr 1cos;1sin22 Vv∞∞θ V (3.7.2) r∂∂θ r rr
In order to locate the stagnation point, assign the velocity components in Eq. (3.7.2) to zero value and simultaneously solve for r and θ . There are two stagnation points,
located at (rR,θπ) = ( ,0) and( R , ) and denoted by points A and B, respectively. The equation of streamlines that passes through the stagnation points A and B, is given by the following expression;
R2 ψθ= −= Vr∞ sin 12 0 (3.7.3) r
This equation is satisfied by rR= for all values of θ . Since R is a constant, Eq. (3.7.3) may be interpreted as the equation of a circle with radius R with center at the
origin. It is satisfied by θπ= 0 and for all values of R . Different values of R may be obtained by varying the uniform velocity and/or doublet strength. Hence, entire horizontal axis through the points A and B, extending infinitely far upstream and downstream, is a part of stagnation streamline. The above discussions can be summarized as follows;
• The dividing streamline ψ = 0 that passes through the stagnation points A and B as shown in Fig. 3.7.1. • The dividing streamline is a circle of radius R . The family of circles can be
obtained by assigning different values of R with various doublet strength and free stream velocity. • The flow inside the circle is generated from the doublet whereas flow outside the circle comes from the uniform flow. So, the flow inside the circle may be replaced by solid body and the external flow will not feel the difference. • Thus, the inviscid, irrotational, incompressible flow over a circular cylinder of
radius R can be simulated by adding a uniform flow of velocity V∞ and a Λ Λ doublet of strength and R is related toV∞ and . Λ R = (3.7.4) 2πV∞
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Referring to the Fig. 3.7.1, it is seen that the entire flow field is symmetrical about both horizontal and vertical axes through the center of the cylinder. It means the pressure distribution is also symmetrical about both the axes. When the pressure distribution over the top part of the cylinder is exactly balanced by the bottom part, there is no lift. Similarly, when the pressure distribution on the front part of the cylinder is exactly balanced by rear portion, then there is no drag. This is in contrast to the realistic situation i.e. a generic body placed in a flow field will experience finite drag and zero lift may be possible. This paradox between the theoretical result of zero drag in an inviscid flow and the knowledge of finite drag in real flow situation is known as d’ Alembert’s paradox.
Pressure Coefficient
In general, pressure is a dimensional quantity. Many a times, it is expressed in a non- dimensional form with respect to free stream flow and the ‘pressure coefficient’ is defined as follows;
pp−−∞∞ pp cp = = (3.7.5) q∞ 1 2 ρ∞∞V 2
Here, ρ∞∞and V are the free stream density and free stream velocity, respectively. The
term q∞ is called as dynamic pressure. For incompressible flow, if a body is immersed in the free stream, then Bernoulli’s equation can be written at any arbitrary point in the flow field as,
11 p+=+ρρ Vp22 V ∞22 ∞∞
1 22 ⇒−pp∞∞ =ρ ( V − V) (3.7.6) 2 2 pp− ∞ V ⇒=cp =−1 qV∞∞
On the surface of the cylinder (rR= ) shown in Fig. 3.7.1, the velocity distribution can be obtained from Eq. (3.7.2) i.e.
vr =0; vVθ = − 2∞ sinθ (3.7.7)
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Fig. 3.7.2: Maximum velocity in the flow over a circular cylinder.
Here, vr is geometrically normal to the surface and vθ is tangential to the surface of
the cylinder as shown in Fig. 3.7.2(a). The negative sign signifies that vθ is positive
in the direction of increasing θ . It may be observed that the velocity at the surface
reaches to maximum value of 2V∞ at the top and bottom of the cylinder as shown in Fig. 3.7.2(b). Eqs. (3.7.6) and (3.7.7) can be combined to obtain the surface pressure coefficient as,
2 cp =1 − 4sin θ (3.7.8)
The pressure distribution over the cylinder is plotted in Fig. 3.7.3. Here, cp varies from 1 at the stagnation point to -3 at the points of maximum velocity. It is also clear that the pressure distribution at the top half the cylinder is equal to the bottom half and hence the lift is zero. Similarly, the pressure distribution on the front part of the cylinder is exactly balanced by rear portion and there is no drag. Both, normal force
and axial force coefficients (ccnaand ) are same as lift and drag coefficients. They are calculated from cp as given below;
11c TE ==−===−= cl c n ∫∫( cpl,, c pu) dx0; cd c a ( cpu,, c pl) dy 0 (3.7.9) cc0 LE
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Here, LE and TE stands for leading edge and trailing edge, respectively. The subscripts u and l refers to upper and lower surface of the cylinder. The chord c is the diameter of the cylinder (R) .
Fig. 3.7.3: Surface pressure coefficient for a circular cylinder.
When the surface pressure matches with free stream pressure, then Eq. (3.7.8) reduces to,
1 c =−1 4sin2θθθ = 0 ⇒ sin =±⇒=3000 , 150 , 210 0 , 330 0 (3.7.10) p 2
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These points as well as the stagnation points and location of minimum pressure are illustrated in Fig. 3.7.4.
Fig. 3.7.4: Pressure values at various locations on the surface of the cylinder.
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Module 3 : Lecture 8 INVISCID INCOMPRESSIBLE FLOW (Superposition of Potential Flows - III)
Lifting Flow over a Circular Cylinder
When a doublet flow is superimposed on a uniform flow, the combined flow fields can be visualized as possible flow pattern over a circular cylinder. In addition, both lift and drag force are zero for such flows. However, there are other possible flow patterns around a circular cylinder resulting non-zero lift. Such lifting flows are discussed here.
Consider the flow synthesized by addition of the non-lifting flow over a cylinder and a vortex of strength Γ as shown in Fig. 3.8.1. The stream function of for a circular cylinder of radius R is given by the following equation.
R2 Λ ψθ1 =Vr∞ sin 1 −= ; R (3.8.1) 2 π rV2 ∞
Fig. 3.8.1: Superposition of non-lifting flow over a cylinder and a vortex.
As discussed in the previous lecture, different values of R can be obtained by assigning the various values of doublet strength (Λ) and uniform free stream velocity
(V∞ ) to synthesize the flow over a circular cylinder. Now, the stream function for a vortex of strength Γ may be written as,
Γ ΓΓ Γr ψ 21=lnrc += ln r − ln R = ln (3.8.2) 2π 22 ππ 2 πR
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Γ Since c1 is any arbitrary constant, it can be replaced with − ln R in the Eq. 2π (3.8.2). The resulting stream function for the flow pattern is given by the sum of the stream functions, i.e.
Rr2 Γ ψψ=+= ψ θ − + 12Vr∞ sin 12 ln (3.8.3) rR2π
The streamlines expressed by Eq. (3.8.3), represents the equation of a circle of radius R . A special case will arise that will represent the flow over a circular cylinder when Γ=0 . If rR= thenψ = 0 for all values of θ . The velocity fields can be obtained by differentiating Eq. (3.8.3) i.e.
1 ∂ψ R2 = = − θ vVr 12 ∞ cos rr∂θ (3.8.4) ∂Γψ R2 =−=−+θ − vVθ 12 ∞ sin ∂ rr 2π r
In order to locate the stagnation points, one can put vvr =θ = 0 in Eq. (3.8.4) and
solve for resulting coordinates (r ,θ ) :
Γ rR=⇒=−sinθ (3.8.5) 4πVR∞
ππ Since Γ is a positive number, the value of θ must be lie in the range and 2 and there are three possibilities;
Γ Case I: If <1 , then the two stagnation points are shown by the points ‘A π 4 VR∞ and B’ lies in the bottom half of the cylinder as shown in Fig. 3.8.2. The locations of these points are given by the Eq. (3.8.5).
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Γ Case II: If =1 , then there is one stagnation point on the surface of the π 4 VR∞ cylinder at the point ‘C’ as shown in Fig. 3.8.2. It means that the point ‘A and B’ π come closer meet at point ‘C’ on the surface at rR=and θ = − . 2
Γ Case III: When, >1, no interpretation can be made from Eq. (3.8.5). 4πVR∞
Referring to Eq. (3.8.4), the stagnation point vr = 0 is satisfied for both ππ π rR=andθ = or − . Now, substitute θ = − in Eq. (3.8.4) and solve for r by 22 2 setting vθ = 0 at the stagnation point.
2 ΓΓ rR=±−2 (3.8.6) 44ππVV∞∞
Eq. (3.8.6) is a quadratic equation and the two possible solutions can be interpreted as stagnation points: one lies inside the cylinder (point ‘D’) and other lies outside the cylinder (point ‘E’) as shown in Fig. 3.8.2. Physically, the point ‘D’ is generated within the cylinder rR< , when a doublet flow at origin is superimposed on a vortex flow while the point ‘E’ lies on same vertical axis for rR> .
Fig. 3.8.2: Stagnation points for lifting flow over a cylinder.
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Based on the results shown in Fig. 3.8.2, the following inferences can be made;
• The circular streamline ψ =0 and rR = is one of the allowed streamline in the synthesized flow field that divides the doublet flow and vortex flow. So, one can replace it as a solid body i.e. circular cylinder and the external flow will not feel the difference. The free stream can be considered as a vortex flow. • With reference to the solid body, the stagnation point ‘D’ has no meaning and only point ‘E” is the meaningful stagnation point. • Since, the parameter Γ can be chosen freely, there are infinite numbers of possible potential flow solutions, for incompressible flow over a circular cylinder. This is also true for incompressible potential flows over all smooth two-dimensional bodies.
Lift and Drag Coefficients for Circular Cylinder
Intuitively, one can say that there is a finite normal force when a circular cylinder is placed in a vortex flow while the drag is zero i.e. d’Alembert’s paradox still prevails. Let us quantify the results;
First, the velocity on the surface of the cylinder (rR= )can be written as,
Γ Vv==−−2 V sinθ (3.8.7) θ ∞ 2π R
The pressure coefficient is obtained as,
222 V Γ 2 2 ΓΓ sinθ cp =−1 =−− 1 2sinθθ − =−1 4sin + + π ππ V∞ 22RV∞ RV∞∞ RV (3.8.8)
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The force coefficients can be obtained by integrating pressure coefficient and skin friction coefficient over the surface. For the inviscid flow, there is no skin friction coefficient. Hence, the drag coefficient is written as,
1TE 11TE TE = −= − cd ∫( cpl,, c pu) dy ∫∫ cpu, dy cpl, dy (3.8.9) 2RLE 22 RRLE LE
Converting Eq. (3.8.9) to polar coordinates by replacing yR=sinθ ; dyR = cos θθ d, we can obtain,
1102π = θθ− θθ ccd∫∫ pu,,cos d cplcos d (3.8.10) 22ππ
Here, the first part of integration is performed from the leading edge (i.e. front point) and moving over the top surface. In the second part, the integration is done from the leading edge moving over the bottom portion of the cylinder. Finally, Eq. (3.8.10) can be written as,
1102ππ 1 2 =−−θθ θθ =−θθ cdp∫∫ cdcos cd pcos ∫ cdpcos (3.8.11) 22ππ 20
Substitute the value of cp from Eq.(3.8.8) in Eq.(3.8.11),
2π 2 1 2 2ΓΓ sinθ cdd =−−+1 4sin θ +cosθθ (3.8.12) 22∫ ππRV RV 0 ∞∞
Use the following trigonometric relations in Eq. (3.8.12);
22ππ 2π ∫∫cosθ= 0; sincos θ θθdd= 0; ∫ sin2 θ cos θθ= 0 (3.8.13) 00 0
It leads to cd = 0 , which implies that the drag on a cylinder in an inviscid, incompressible flow is zero, regardless of whether or not the flow has circulation about the cylinder. The lift can be evaluated in the similar manner from the first principle i.e.
12R 1122RR = −= − cl ∫( cpl,, c pu) dx ∫∫ cpl, dx cpu, dx (3.8.14) 2R0 22 RR00
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Converting Eq. (3.8.14) to polar coordinates by replacing x= Rcosθ ; dx = − R sin θθ d , we can obtain,
1120ππ 1 2 =−+=θθ θθ− θθ cl ∫∫ cpl,,sin d cpusin d ∫ cdpsin (3.8.15) 22ππ 20
Substitute the value of cp from Eq.(3.8.8) in Eq.(3.8.15),
2π 2 1 2 2ΓΓ sinθ cdl =−−+1 4sin θ +sinθθ (3.8.16) 22∫ ππRV RV 0 ∞∞
Use the following trigonometric relations in Eq. (3.8.16);
22ππ 2 π ∫∫sinθ= 0; sin23 θθdd= π ; ∫ sin θθ= 0 (3.8.17) 00 0
The lift coefficient can be obtained as,
ΓΓL′ cl =⇒= (3.8.18) RV∞∞1 2 RV ρ∞∞VS 2
The value of lift per unit span (L′) can be obtained by considering the plan-form area
SR= 2 in Eq. (3.8.18) and after simplification, one can obtain,
LV′ =ρ∞∞ Γ (3.8.19)
It is seen from Eq.(3.8.19) that the lift per unit span for a circular cylinder in a given free stream flow is directly proportional to the circulation. This simple and powerful relation is known as Kutta-Joukowski theorem. This result shows the importance of the concept circulation and the same result can be extended for two-dimensional bodies. The inviscid potential flow does not provide proper explanation for drag calculation because zero drag in a flow field is quite un-realistic. Because of viscous effects the flow separates from the rear part of the cylinder, creating recirculating flow in the wake downstream of the body. This separated flow greatly contributes the finite drag measured for the cylinder. However, the prediction of lift by Kutta-Joukowski theorem is quite realistic.
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Magnus Effect
The general idea of generation of lift for a spinning circular cylinder can be extended to sphere. Here, non-symmetric flows are generated due to spinning of bodies in all dimensions. It leads to the generation of aerodynamic force perpendicular to the body’s angular velocity vector. This phenomenon is called as Magnus effect. The typical examples include the spinning of three-dimensional object such as soccer, tennis and golf balls where the side force is experienced.
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