Department of Aeronautics Imperial College London MSc in Advanced Computational Methods for Aeronautics and MSc in Advanced Aeronautical Engineering Introduction to Fluid Dynamics Self-study module Prof. Sergei Chernyshenko and Dr. Kevin Gouder For 2020-2021 Copyright notice. This text has a significant overlap with the content of Flow Illustrator website (currently at http://www.flowillustrator.com/). The copyright of this part of the content is determined by the copyright notices on that website. The rest is copyright of Imperial College London.
Transcript
Department of Aeronautics
Imperial College London
MSc in Advanced ComputationalMethods for Aeronautics and MSc inAdvanced Aeronautical Engineering
Introduction to Fluid Dynamics
Self-study module
Prof. Sergei Chernyshenko and Dr. Kevin Gouder
For 2020-2021
Copyright notice. This text has a significant overlap with the content of Flow
Illustrator website (currently at http://www.flowillustrator.com/). The copyright
of this part of the content is determined by the copyright notices on that website.
The rest is copyright of Imperial College London.
THIS PAGE IS INTENTIONALLY LEFT BLANK
Preface
This module is a part of the MSc courses at the department ofAeronautics of Imperial College London. At the time of writ-ing the relevant courses are “Advanced Computational Methodsfor Aeronautics, Flow Management and Fluid-Structure Interac-tion” and “Advanced Aeronautical Engineering”. The module,called “Introduction to Fluid Dynamics”, is compulsory, but notexamined. Since 2016-2017 this module is an online self-taughtcourse. It requires about 20 hours of work from the student.The students are expected to complete the module before ar-riving to College. At the start of the course there will be tworevision seminars for this module, in which the lecturers willanswer questions.
The main goal of the module is to bring the students to a morecommon background level. By their very nature the MSc coursesare advanced, which implies a substantial prior knowledge. Bydesign, however, the courses are aimed to serve not only thosewishing to continue their study of aerodynamics and aeronau-tics, but also the discipline-hoppers: students whose first degreewas in related but different disciplines, provided that they havea sufficient mathematical background. Another reason for thediversity of the student backgrounds is their origins: our stu-dents come from around the world, and the educational systemsdiffer.
One distinguished feature of the approach to education inthe United Kingdom is that subjects are taught in a repetitivemanner, which each new module revisiting the already coveredmaterial but at a new level, with greater depth and detail, oftenlooked at from a new viewpoint, and taught more intensely. Inthe MSc course this comes as a surprise to many of the overseasstudents. In the MSc courses at the department of Aeronautics,general fluid dynamics is taught in the Introduction to FluidDynamics module (that is this one), then again in the Funda-mentals of Fluid Mechanics module, which is the only examinedmodule taught at what essentially is an undergraduate level,
and then again as part of the modules on Compressible Flow,Hydrodynamic Stability, Separated Flows and Fluid-StructureInteraction, Navier Stokes Equations and Turbulence Modelling,and to a lesser extend in several other modules. To get the fullbenefit from the course, students are advised to keep in mindthis characteristic feature of the approach to teaching fluid dy-namics and, in fact, other subjects as well.
Hence, this introductory module is only a start. It has a sig-nificant overlap with the “Foundations” subsection of the “FluidDynamics” part of the Flow Illustrator website (at the time ofwriting at http://www.flowillustrator.com), which is aimed atgeneral public and is less mathematical. The students are ad-vised to check it before reading further, to see if the style and thematerials of this website fit better their preferences. After thecompletion of this module the student should find the contentof the Flow Illustrator website very easy to understand.
When thinking about fluid flows, fluid dynamicists use specialterminology: velocity field and pressure field, fluid particles, ma-terial lines, streamlines and streak lines, vorticity, Eulerian andLagrangian viewpoints and many more. These terms constitutethe language, the basic words of which need to be understoodfirst. We will explain a few of these notions in this section.While it is expected that the student is familiar with the vectorcalculus, explanations are always given. Nevertheless, it mightbe useful to have a look at the Appendix A explaining vectornotation.
1.1 Continuum
Fluid, as almost all other things, consists of molecules separatedby empty space. Usually the mean free path, that is the averagedistance the molecules travel between collisions, is much smallerthan the characteristic dimensions of the flow. Then it is conve-nient to imagine that the fluid mass is continuously distributedin space. This approach is common and is called continuummechanics.
1.2 Characteristic dimensions or characteristic scales
Characteristic dimensions and scales are the relevant dimensionsand scales. Knowing characteristic scales gives meaning to thewords ‘small’ and ‘large’. Is 1 mm small? This depends on whatit is to be compared with. Compared to the size of a car 1 mmis very small, but compared to the size of a virus it is very large.And compared to the size of a rain drop 1 mm is neither largenor small.
For example, for an aircraft the characteristic length can betaken as the length of the aircraft, and the characteristic veloc-ity can be taken as the flight velocity. The characteristic timecan be obtained as the characteristic length divided by the char-acteristic velocity. The exact values of the characteristic scales
1.3 Fluid particle 9
can be selected with a certain freedom provided that they re-main physically relevant to the flow in question. The Reynoldsnumber, defined in Section 2.1, becomes physically meaningfulwhen it is calculated with characteristic scales.
1.3 Fluid particle
Considering fluid as a continuum allows us to choose an arbitrarypoint in fluid and speak about fluid properties at this point,such as velocity and pressure. If the point keeps moving withthe fluid velocity at this (moving) point then we call it a fluidparticle. A fluid particle is, therefore, a model of a fluid volumevery small as compared to the characteristic flow dimension (sothat the size of this volume can be neglected) but very large ascompared to the free path.
1.4 Material line
Let us imagine a curve in space and mark all the fluid particleson this curve. The particles can move with time and so will dothe curve formed by these particles. Such a curve moving withthe fluid is called a material line. Fluid motion can translatematerial line, rotate it, stretch and bend it, so that the motionof a material line can be quite complicated.
1.5 Velocity field
Every fluid particle has its own velocity. Since fluid particles fillthe entire space occupied by the fluid, fluid velocity is a field. Itis often denoted as u(x), to emphasise that the velocity vectoru depends on the position of the fluid particle. The position isdetermined by the vector x, which is a vector going from thecoordinate origin to the point in question. Vectors are denotedby bold letters.
10 1 THE LANGUAGE OF FLUID MECHANICS
1.6 Streamlines
To visualise the flow one can try to select a number of points init, and at each point to draw the velocity vector of the corre-sponding fluid particle. Often, however, the result is difficult tounderstand. Instead, one can draw a set of curves along the di-rection of velocity. Such curves are called streamlines. Figure 1illustrates the advantage of visualising flows with streamlines.It shows a sketch of a separated flow past a circular cylinder atthe value of the Reynolds number Re of about 20.
Figure 1: separated flow past a circular cylinder at the value of the Reynoldsnumber Re of about 20.
1.7 Planar and two-dimensional flows
The flow in figure 1 is shown as a flat image. Strictly speakingthis is only possible if the velocity vectors lay in the plane of thepicture, and it also implies that in the real three-dimensionalworld the velocity is the same at all points along the lines per-pendicular to the plane of the picture. This is why the flow in thepicture is the flow past a circular cylinder: what is shown is onlyone cross-section of the flow, and this is enough since it wouldbe the same in all other cross-sections. Planar two-dimensionalflows can only be an approximation of the real world, since noobject can be infinitely long in the direction perpendicular tothe plane of the picture.
1.8 Steady and unsteady flows 11
1.8 Steady and unsteady flows
At Re=20 the flow past a circular cylinder is steady. This meansthat the velocity field does not change in time. In steady flowsfluid particles move along the streamlines. However, if the flowis unsteady then the streamlines, too, vary with time, and as aresult the trajectory of a fluid particle can be different from theinstantaneous streamline the particle is on at a particular mo-ment. Unsteady velocity field can be denoted as u(t,x), wheret stands for time.
1.9 Lagrangian and Eulerian viewpoints
Imagine two people, Mr. Euler and Mr. Lagrange, makingtemperature observations in order to calculate the rate at whichthe temperature is varying. Mr. Euler is standing on a bridge,while Mr. Lagrange is in a boat carried by a river. When theboat passes the bridge, they discover that their observed ratesdiffer. Why?
This happens because the temperature differs in different lo-cations on the ground. To calculate the rate, they divide thedifference between the two observed temperatures by the timeinterval between the measurements. Even if one of the mea-surements was made when the boat was passing the bridge, theother measurement was when the boat was away from the bridge.Then, even if they took this other measurements at the same in-stant, the results can differ.
In the example with the boat it was assumed that the boatwas carried by the fluid, that is, it was moving with the velocityof the fluid itself. The Eulerian viewpoint consists in consider-ing quantities as dependent on time and point in space. TheLagrangian viewpoint consists in considering quantities as de-pendent on time and the fluid particle. Lagrangian viewpointis particularly useful when the laws of fluid motion are derivedfrom the Newton laws. Once derived, however, these laws aremore convenient to use from the Eulerian viewpoint.
We will now introduce a simple notation, which will be conve-
12 1 THE LANGUAGE OF FLUID MECHANICS
nient in what follows. In general, the temperature T of the fluidvaries from point to point and is therefore a field: T = T (t,x).The rate of change is a derivative; in this case it is the derivativewith respect to time. As we just discussed, at the same pointx we can have two different values of time derivative, the oneobtained from the Eulerian viewpoint and the other obtainedfrom the Lagrangian viewpoint. The Lagrangian derivative isdenoted in the standard way: for example, the time derivativeof temperature T is denoted dT/dt. The Eulerian, also calledlocal, derivative is denoted similarly, but using a special form ofthe letter d : ∂T/∂t.
If ∂T/∂t = 0 then the field T is independent of time, so thatwe can say that T = T (x). Such fields are called steady.
If dT/dt = 0 then T at each fluid particle is independent oftime. Such fields are called frozen. One can imagine that thevalues of the field quantities were glued by frost to fluid particlesand move with them.
Finally, the difference between these two derivatives is calleda convective derivative and denoted as u · ∇T , where u is thevelocity field of the fluid particles, and ∇ is a special symbolcalled nabla. This notation is introduced such that the materialderivative is a sum of the local derivative and the convectivederivatives:
dT
dt=∂T
∂t+ u · ∇T.
Of course, the rate of change, or derivative, can be calculatedfor other quantities, for example, for the velocity itself, and then
du
dt=∂u
∂t+ u · ∇u.
If no forces were acting on fluid particles then, accordingNewton’s first law of motion, each particle would move withconstant speed, and the material derivative of the velocity wouldbe zero: du/dt = 0. The local derivative, however, would not benecessarily zero, since we would have
1.10 Activity 13
∂u
∂t= −u · ∇u.
The above equation means, therefore, that the velocity ischanging at a fixed point in space only because fluid particleswith different velocities are passing through this point of space,while being carried, or advected, by the fluid.
In reality, of course, there are forces acting on the fluid par-ticles, and these forces will be discussed in the following pages.
1.10 Activity
In movies made with Flow Illustrator randomly chosen fluid par-ticles are shown as white dots. Watch a movie from the Galleryand observe the motion of the fluid particles. Decide which flowsshown are steady and which are not.
14 2 PHENOMENOLOGY OF FLUID FLOWS
2 Phenomenology of fluid flows
In this section only a general outline of the main fluid flow fea-tures is given.
2.1 Reynolds number
A fluid flow past the body of the same shape can be of quitedifferent character depending on other parameters, such as thesize of the body, the average flow speed, fluid density and vis-cosity. Remarkably, the observation show and the theory proves(as discussed in Section 7) that fluid flows can be classified interms of their dimensionless Reynolds number which is definedas
Re ≡ DU
ν, (1)
where D is the characteristic size of the body, U is the charac-teristic velocity, and ν is the kinematic viscosity. Depending onthe value of the characteristic Reynolds number, the flow pastthe same body can be steady or unsteady, laminar or turbulent,attached or separated.
For small, that is less than 1, values of Re the flows are typi-cally steady, and more often than not attached. However, flowswith small Re are less frequently encountered in practice. In-deed, consider the flow of air induced by a walking man. Takingthe characteristic dimension of a human as 1 m, the walkingspeed as 5 km/h, and the air kinematic viscosity 1.511·10−5 m/s,which is the value for 20◦C at 1 atm gives Re = 91, 912. SinceRe is proportional to the velocity and the size, for anything ofa larger size or moving in the air faster Re will be even greater.The kinematic viscosity of water is about 10 times smaller thanthat of air, and, hence, for the same size and speed in water Rewould be about 10 times larger. In human-related applications,therefore, large Re are typical, and we will pay more attentionto this case.
2.2 Separation 15
2.2 Separation
Apart from the Reynolds number, the main factor determiningthe nature of the flow is the body shape. Shapes can be stream-lined or bluff. A typical airfoil at a small angle of attack is astreamlined body, but at a very large angle of attack it is a bluffbody. The difference between streamlined and bluff bodies isthat the flows past streamlined bodies remains attached evenat high Re, while flow past bluff bodies become separated asRe increases. So far a rigorous definition of separated flows wasfound only in the special case of high-Reynolds-number asymp-totic theory, which is outside the scope of this course. However,telling a separated flow from a non-separated flow when one seesit is relatively easy. A characteristic feature of separated flowsis flow reversal, that is the presence of a region, called separa-tion eddy, where the fluid moves in the direction opposite to themain direction of the flow. For example, the flow in figure 2 isattached, but the flow in figure 3 is separated.
Figure 2: Streamline of irrotational flow past a circular cylinder, which hasthe same topology but different shape as the streamlines of flow at very smallRe.
Majority of separated flows encountered in practice are un-steady even at relatively small Re. For example, as Re (basedon the diameter) increases beyond 50, the steady flow becomesunstable and evolves into a time-periodic flow, in which the ed-dies detach alternately from the sides of the cylinder and movedownstream forming a structure called a von Karman vortexstreet, see figure 4.
16 2 PHENOMENOLOGY OF FLUID FLOWS
Figure 3: The streamline topology of a steady separated flow past a circularcylinder, 3− 5 < Re < 47.
Figure 4: Von Karman vortex street: a snapshot from the video of the flowpast a circular cylinder at Re = 100. (Made at www.flowillustrator.com.Flow Illustrator calculates Re based on the height of the image. Here it wasrecalculated for the value based on the diameter.)
Figure 5: An airfoil and a cylinder (yes, yes, that small black dot on top isit) of equal drag at high Re.
Separation is an important phenomenon. Figure 5 shows anairfoil and a circular cylinder which experience equal drag atvery high Reynolds numbers. The huge difference in size is dueto the flow being separated in the cylinder case and attached inthe case of an airfoil. Separation is the subject of the SeparatedFlow and Fluid-Structure Interaction module of our AdvancedComputational Methods MSc course.
2.3 Instability, transition, and turbulence
As the Reynolds number increases steady attached flows alsoloose stability and become unstable. This usually happens at
2.3 Instability, transition, and turbulence 17
much larger Re than in the case of a separated flow. The pri-mary reason for this is the shape of the velocity profile. In gen-eral, flows with velocity profiles without inflection points, as infigure 6(a), are much more stable than the flows with inflectionpoints, as in figures figure 6(b,c). The difference in the criticalvalue of Re at which stability is lost in flows with and withoutinflection points varies, but typically it is more than an orderof magnitude (more than 10 times). Stability of fluid flows isthe subject of the Hydrodynamic Stability module of our MSccourses.
(a)
a
b
(b)
a
b
(c)
a
b
Figure 6: Near-wall velocity profile without an inflection point (a); near-wallvelocity profile with an inflection point (b); separated flow velocity profilealways have an inflection point (c).
In the flow past a circular cylinder for yet higher Re the reg-ular vortex shedding becomes irregular, chaotic, and the spatialpattern of the flow also becomes irregular, disordered, eventuallyresulting in the so-called turbulent flow regime. Turbulent flowsare characterised by simultaneous presence of many temporaland spatial scales, including very small scales, high energy dis-sipation due to the presence of those small scales, and by highsensitivity of the instantaneous flow features. Attached flowsoften do not exhibit a time-periodic behaviour. Instead, if thesteady flow is unstable, the flow becomes turbulent. Turbulentflows can easily be distinguished by the shape of the plots ofvelocity (see figure 7), by the noise they are producing, and bythe irregularity of the flow visualisations. Turbulence is the sub-ject of the Navier Stokes Equations and Turbulence Modellingmodule of our MSc courses.
18 2 PHENOMENOLOGY OF FLUID FLOWS
Figure 7: Time traces of velocity in (a) laminar and (b) turbulent flow, asseen by a hot-wire probe.
2.4 Boundary layers
Formation of boundary layers at high Re is one of the mostimportant features of fluid flows. From the physical viewpoint,Reynolds number is the ratio of the average magnitudes of iner-tia and viscous friction in the bulk of the flow. If Re is high onecould, therefore, expect to be able to neglect the friction insidethe fluid. However, without friction the fluid would slide alongthe solid walls bounding it, while in reality the fluid velocity isalways equal to the velocity of the solid at their interface. Thus,at high Re a thin layer of fluid adjacent to the solid wall devel-ops high velocity gradients, with the velocity varying from thewall velocity at the wall to the velocity in the bulk of the flow.This thin layer is called a boundary layer. Boundary layers canbe laminar or turbulent, and they can separate from the surfaceand continue into the bulk of the fluid in the form of so-calledmixing layers. Discussion of boundary layers will be continuedonce the quantity called vorticity will be introduced. Variousaspect of the boundary layer concept are also discussed in manymodules of our MSc course.
2.5 Drag of various bodies
In conclusion of this overview, we give also a summary plotof drag of various bluff and streamlined bodies (figure 8). Forinterpreting this plot it is convenient to assume that the flow atinfinity and the body size are fixed. Then the drag coefficientis proportional to the drag, and an increase in Re correspondsto a decrease in viscosity. Note the logarithmic scales and notethe range of Re displayed.
The bottom curve in figure 8 is for a thin flat plate parallel
2.5 Drag of various bodies 19
Flat plate D
DCircle
0.5D
Ellipse
D
Airfoil
Flat plate 0.18D
1
0.1
0.01
CD
104 105 106 107
Re = UD/ν
Figure 8: Drag coefficients of bluff and streamlined bodies.
to the main flow direction. For smaller Re the flow is laminar,and decrease in viscosity leads to a decrease in the drag, as itcan be expected. At Re somewhat below 106 a transition toturbulence occur. The drag then increases with a decrease inviscosity (increase in Re) until the transition effect saturates.After that, further decrease in viscosity reduces the drag, but ata much slower rate than for the laminar flow.
The top curve in figure 8 is for a thin flat plate perpendicularto the main flow direction. In this case the sharp edges of theplate cause flow separation, which is a very powerful featuredominating the flow. The drag is high and independent of Re.
The curve for a circular cylinder (circle) exhibits an interest-ing phenomenon called drag crisis. Somewhere between Re =105 and Re = 106 the drag coefficients drops as Re increases.This happens when the boundary layer on the cylinder surfaceundergoes transition to turbulence. Turbulent boundary layersare more resistant to separation, and as a result the separationregion shrinks, decreasing the drag.
The drag curves of the ellipse and the airfoil have a behaviourintermediate of that for the circle and the flat plate aligned withthe flow.
20 3 CONSERVATION OF MASS AND INCOMPRESSIBILITY
3 Conservation of mass and incompressibility
3.1 Mass conservation
Everyone knows that mass is conserved. What does this meanfor a fluid flow? Consider a branched pipe shown in the figure 9.Suppose that a fluid is flowing through it as shown with arrows.Then mass conservation means that the rate, with which themass of the fluid inside the pipe varies is the difference betweenthe rate with which the fluid carries mass through inlet 1, andthe sum of the rates with which the fluid carries mass throughoutlets 2 and 3.
Figure 9: Branching pipe
The mass of the fluid inside the pipe equals the product ofthe fluid density and the volume of the fluid inside the pipe.Since the volume does not vary with time, the rate with whichthe mass of the fluid inside the pipe varies is proportional to therate with which the density of the fluid inside the pipe varies.If the density variation can be neglected, it is very convenient,because it simplifies the matter.
Fluids of constant density are called incompressible fluids.In particular, for an incompressible fluid we can say that therate with which the fluid carries mass through inlet 1 equalsthe sum of the rates with which the fluid carries mass throughoutlets 2 and 3. This is true even when the flow rates varywith time. The mass flow rate through a cross-section of a pipe
3.1 Mass conservation 21
is the product of the fluid density, average velocity componentperpendicular to the cross-section, and the cross-section area.Using the subscript to denote the quantity at the correspondingcross-section and taking into account that the density is thesame everywhere, mass conservation gives
v1 · A1 = v2 · A2 + v3 · A3,
where A stands for the cross-section area and v is the velocityacross it. This is, indeed, simple.
Under normal conditions density variation of the density ofliquids, such as water, can indeed be neglected. For gases, suchas air, this is more complicated. However, if the variation of theabsolute pressure and the absolute temperature of the gas canbe neglected, then from the gas laws it follows that the densityvariation can also be neglected.
Pressure and density variation can be due to various reasons.For example, air can be pumped into a car tire to the pressure of2 atm, while on the outside the absolute pressure is 1 atm. If nowwe can bleed the air from the tire, the air pressure will changeby the factor of two, and, therefore, in such a flow one cannotassume the air to be incompressible. Similarly, large tempera-ture variation can be caused by direct heating, causing changesin density thus also leading to the violation of incompressibilityassumption.
Variation of pressure and temperature can be caused by theflow itself. The theory of compressible flows shows that therelative magnitude of these variations is of the same order as theMach number squared. The Mach number M is the ratio of thecharacteristic flow velocity to the speed of sound. Under normalconditions in the air the speed of sound is about 340 m/s (orabout 1,200 km/h, or 770 mph). Incompressible flow assumptionis usually considered to be valid for M < 0.3.
Although rare, there are flows in which the free path of themolecules is not small enough as compared to the characteristicdimensions of the flow. This might be the case for spaceshipsin the upper atmosphere and for flows in very thin tubes, as
22 3 CONSERVATION OF MASS AND INCOMPRESSIBILITY
encountered in nanotechnology or medical applications. In thiscase the assumption of continuum media is not valid.
If, however, the fluid can be considered as a continuum, theMach number is less than 0.3, and the relative variation of abso-lute pressure and temperature due to external reasons is smallenough then the fluid flow can be considered as a flow of anincompressible fluid. (Small enough means that the acceptableerror is noticeably greater than the relative variation of absolutepressure and temperature).
3.2 Continuity equation
Mathematically the property of incompressibility of the fluidhaving the velocity field u can be expressed as
∇ · u = 0. (2)
This is called the continuity equation for an incompressible fluid.In Einstein’s notation and Cartesian coordinates it is
∂ui∂xi
= 0.
Without Einstein’s notation, and taking u = (u, v, w), and x =(x, y, z) in Cartesian coordinates it can also be written as
∂u
∂x+∂v
∂y+∂w
∂z= 0. (3)
23
4 Momentum conservation and the friction
law
According to second Newton’s law acceleration of a materialbody is equal to the force acting on the body divided by themass of the body. This is equivalent to the statement that themomentum is conserved. The Navier-Stokes equations are theequations expressing second Newton’s law or, equivalently, mo-mentum conservation, in a viscous fluid.
4.1 Pressure
Pressure and friction are forces acting in fluid. Everyone hasexperienced pressure in the everyday life. When a strong windbreaks down an umbrella, this happens because the pressure onone side of the umbrella cloth is much higher than the pressureon the other side. The pressure difference can be created by theflow of fluid, and it can cause the fluid to flow.
Figure 10: Forces acting on an area
To be precise, pressure is a force per unit area. Imagine asmall surface element inside the fluid (see figure 10). The fluidon one side of this element can exert, though the element, a forceon the fluid on the other side. By Newton’s third law the fluidon the other side will, of course, exert an equal and oppositeforce through the same surface. Pressure is the force divided bythe area of the surface element. Since pressure can change frompoint to point, the element has to be small, so that the variationof pressure over the element could be neglected.
24 4 MOMENTUM CONSERVATION AND THE FRICTION LAW
Pressure has three specific properties, which allow to distin-guish it from friction. First, unlike the friction force, pressurecan be nonzero even if the fluid is at rest. Consider the air insidethe air balloon, for example: the fluid does not move, but thepressure is there. Second, pressure is isotropic. This means thatthe pressure force acts equally in all directions. If the elementin the figure was turned by any angle, the pressure force actingthrough it would be the same. Third, pressure force is alwaysperpendicular to the surface.
4.2 Friction
Another type of force acting inside the fluid is friction. Oureveryday experience suggests that, unlike pressure, friction issmall in air and water. Place the palm of your hand on thesurface of the table and move your hand back and forth. Feelthe friction force. Then lift the hand and move it in the air: theair friction is imperceptible. If so, should one pay attention toit? The answer is a very definite yes. The reason for the largeeffect caused even by a small friction is related to the phenomenaof boundary layers and separation. It will be explained later inthis and other modules of the MSc course.
Figure 11: Velocity profile of a simple shear motion near a wall
We will consider first the simple case when the velocity vec-tors have the same direction everywhere, while the magnitude ofthe velocity varies linearly in the direction perpendicular to thedirection of the velocity. This frequently happens in the vicinityof a wall as is illustrated in figure 11, showing the velocity pro-file of a simple shear motion near a wall. Such a flow is called
4.3 Diffusion of velocity as the effect of viscosity 25
a shear flow because of the shearing motion of the fluid layerswhich slide past one another.
In this case the friction force exerted on the wall will be pro-portional to the shear rate, which in this case can be calculatedas the derivative of u with respect to y, that is du/dy, and sincethe velocity profile is linear, and the velocity at the wall is zero,this is simply u(y)/y. The force τ exerted on the unit area ofthe wall is given by the formula
τ = µ du/dy, (4)
where the constant µ, called the viscosity coefficient, depends onthe fluid. Its numerical value can be found in handbooks, andit usually depends on the temperature. The above formula iscalled Newton’s viscous friction law. The same formula remainsvalid inside the fluid. In that case it gives the force exerted onone another by the layers of moving fluid.
4.3 Diffusion of velocity as the effect of viscosity
Imagine a drop of ink or milk in otherwise clear water at rest,so that the water velocity and the ink velocity are both zero.Everyday experience suggests that the ink will gradually spreadover the volume. This process is called diffusion. Due to thediffusion, the colour of the ink spot will become more and morepale, the size of the spot will increase, while the boundary of thespot will become more and more smeared. Temperature behavesin a similar way in a heat-conducting medium. Diffusion doesnot change the total amount of ink: rather, the ink is being re-distributed between fluid particles. This conservative propertyof the diffusion process is important and is worth remember-ing. The direction of diffusion is from the areas with high inkconcentration to the areas with small concentration. For tem-perature, the diffusion is from hot to cold particles, so that italways tries to make the distribution uniform. Diffusion is intu-itively familiar to most people, and it is also easy to understandand learn.
26 4 MOMENTUM CONSERVATION AND THE FRICTION LAW
The short story is that in an incompressible fluid with con-stant viscosity the action of viscosity is equivalent to diffusionof velocity. In Cartesian coordinates each component of the ve-locity vector is diffused independently. The rate of the diffusiondepends on the fluid properties. The same applies for vortic-ity: the action of viscosity is equivalent to diffusion of vorticity.Each component of vorticity is diffused independently.
The result of diffusion is the variation in time of the valueof the diffused quantity. If, for example, C(t,x) is the scalarfield representing the distribution of the ink concentration then,due to diffusion, it will change in time with the rate which de-pends on the distribution itself. This rate as a function of thedistribution has a special notation, k∆C(t,x), where k is a con-stant determined by the properties of the fluid and the ink, and∆C(t,x) is called a Laplacian of C(t,x). In Cartesian coordi-nates
∆C =∂2C
∂xi∂xi=∂2C
∂x2 +∂2C
∂y2 +∂2C
∂z2 .
4.4 The Navier-Stokes equations
Pressure and friction are the two main forces acting in the fluid,and their effects should be added together to calculate the ac-celeration of the fluid particles by second Newton’s law. It iseasier to remember the result if it is written in the form of asimple formula, which is called the Navier-Stokes equation (inLagrangian frame of reference):
ρdu
dt= −∇p+ µ∆u.
Here u stands for fluid velocity, p for pressure, ρ for density,t for time and µ for dynamic viscosity. The values of µ andρ depend on the fluid and can be found in handbooks. Theformula means:
density × acceleration = pressure force + friction force.
Note that this is a vector equation, since u is a vector field,and so are the other terms. In Cartesian coordinates the Lapla-
4.4 The Navier-Stokes equations 27
cian of the vector field equals the vector field of Laplacians inthe sense that if u = (u, v, w) then ∆u = (∆u,∆v,∆w). Thisis not so in other coordinates systems. In Cartesian coordinates
the gradient of pressure ∇p =(∂p∂x,∂p∂y,∂p∂z
). Using Cartesian
coordinates it is easy to check that ∇ · (∇q(x)) = ∆q(x) forany field q(x). For this reason instead of ∆u one can write ∇2u.The symbol ∆ is often used also for denoting an increment of aquantity. One needs to understand which of the two meaningsis implied from the context.
If the viscosity is not the same at all points in the fluid, or ifthe fluid is not incompressible, the viscous term has a different,more complicated form. Forgetting this is a common source oferrors. Viscosity can be not constant due to temperature varia-tion. Turbulent viscosity, used to model the effect of turbulence,is also not constant.
It is common to divide this equation by density, to obtain
du
dt= −1
ρ∇p+ ν∆u,
where ν = µ/ρ is called kinematic viscosity.
This expresses the law of fluid motion from the Lagrangianviewpoint. That is, du/dt is the rate of the velocity variation asobserved from a particle moving with the fluid. The Euler view-point is often more convenient. Switching to Euler viewpointwill result in a different way of expressing the rate of velocityvariation. It leads to
∂u
∂t= −u · ∇u− 1
ρ∇p+ ν∆u. (5)
This is the Navier-Stokes equation in Euler’s frame of refer-ence. In words, this can be described in the following way. Ata fixed point in space, the velocity changes with time with therate ∂u/∂t for three reasons. First, fluid particles come to thispoint and bring their own velocity there. If it is different fromthe velocity at the particle that occupied the same point in spaceshortly before that, the velocity at that point is changing with
28 4 MOMENTUM CONSERVATION AND THE FRICTION LAW
the rate −u · ∇u. Then there is pressure. If pressure on oneside of the fluid volume is greater than the pressure on the otherside of the fluid volume, the fluid will accelerate, and the corre-sponding rate of the velocity change is denoted −1
ρ∇p. Finally,friction results in the diffusion of velocity, which is expressed byν∆u.
Taking u = (u, v, w), and x = (x, y, z) in Cartesian coordi-nates it can also be written as
∂u
∂t+ u
∂u
∂x+ v
∂u
∂y+ w
∂u
∂z= −1
ρ
∂p
∂x+ ν∆u, (6)
∂v
∂t+ u
∂v
∂x+ v
∂v
∂y+ w
∂v
∂z= −1
ρ
∂p
∂y+ ν∆v, (7)
∂w
∂t+ u
∂w
∂x+ v
∂w
∂y+ w
∂w
∂z= −1
ρ
∂p
∂z+ ν∆w. (8)
29
5 Boundary and initial conditions
5.1 Time marching
Suppose that the fluid fills the entire space, and that we knowthe velocity and pressure distribution at a certain time instantand in the entire space. Can we calculate what the velocity fieldwill be like a little time later? The answer is yes. Indeed, sincewe know the velocity and the pressure, we can calculate thespatial derivatives entering the right-hand side of the Navier-Stokes equations, and then we can calculate the time derivativeof the velocity. We then can use the Taylor expansion withrespect to time to calculate approximately the velocity after ashort time interval ∆t :
u(t+ ∆t,x) ≈ u(t,x) +∂u(t,x)
∂t∆t
at every point x, and the smaller ∆t, the more accurate ourresult is.
If the fluid fills the entire space then nothing else is required.However, if the region occupied by fluid has boundaries then fortime-marching one also needs to know what happens at thoseboundaries, namely, to know what the velocity is at the bound-ary at all times. This cannot be determined from time-marching.Instead, it has to be prescribed.
5.2 Boundary conditions
The most common boundary is the surface of a solid body. Thefluid cannot penetrate the surface of the body, which limits whatthe fluid velocity at the surface can be. The requirement thatthe wall-normal (that is perpendicular to the body surface) com-ponent of the fluid velocity equals the wall-normal componentof the solid wall velocity is called an impermeability condition.Due to the friction between the surface and the fluid, at thesurface the fluid velocity is simply equal to the velocity of thesolid surface. This requirement is called a no-slip boundary con-ditions. In particular, if the body does not move (say, a building
30 5 BOUNDARY AND INITIAL CONDITIONS
in a wind), then at the surface the fluid velocity is zero.
Another common case of a boundary is an imaginary bound-ary, which we add simply because we know what the velocityat that boundary is. For example, for the case of an aircraftflying in an otherwise still air, we can imagine a box around theaircraft, so large that on the surface of the box the air velocityis zero. Then we will not need to think about the fluid motionoutside the box, which makes it easier.
5.3 The pressure problem
Knowing the velocity a little time later, we can try to repeat thisreasoning again and again, stepping in time, thus finding theflow at all later times in a time-marching procedure. However,for this we also need to know the pressure at the new time.Unfortunately, there is no simple way to determine the timederivative of the pressure.
The physics determining the pressure evolution is compli-cated, at least in the general case. It happens that the advectionof velocity (the term −u ·∇u) in the Navier-Stokes equation (5),acting alone, can result in the new velocity field that does notsatisfy the continuity equation (2), and for an incompressiblefluid this violates the mass conservation law. The pressure cor-rects this. For a given velocity distribution at a certain instantthe pressure distribution will be such that at the following in-stant the velocity distribution continues to satisfy mass conser-vation for incompressible fluid. The important observation isthat to start the time-marching procedure one does not need toprescribe the initial distribution of pressure: the initial velocityfield is enough. Note also that changing the pressure by a valuethat is independent of spatial coordinates does not change ∇p.Hence, prescribing boundary and initial conditions only for ve-locity allows to determine pressure up to an uncertain additivefunction of time. To eliminate this uncertainty and thus deter-mine the pressure uniquely one needs to prescribe the value ofpressure at one single point.
The summary of this is: the fluid motion is defined by the
5.3 The pressure problem 31
initial conditions for velocity, and the boundary conditions forvelocity. No initial or boundary conditions for pressure are re-quired, apart from the value at a single point, changing thisvalue would not affect the velocity. Initially, all the velocitycomponents should be given, and as the time progresses, all ve-locity components need to be given at the boundary of the flowregion.
The condition that at any given time the pressure is suchthat the velocity continues to satisfy the mass conservation asthe time varies is hard to use at the intuitive level. It is alsodifficult to use in theoretical analysis and numerical calculations.The two common approaches to overcome this difficulty is theso-called velocity-vorticity formulation and the boundary layertheory, both of which will be explained later in the course.
32 6 UNIDIRECTIONAL STEADY FLOWS
6 Unidirectional steady flows
If, in Cartesian coordinates, only one velocity component is non-zero, the Navier-Stokes equations simplify significantly. Suchflows are called unidirectional. Let u 6= 0 but v = 0 and w = 0,and let us limit the considerations to two-dimensional steadyflows, so that the derivatives with respect to z and the time arezero. Then the continuity equation (3) becomes
∂u
∂x= 0,
which implies that u depends only on y but not on x. The secondmomentum equation (7) becomes
0 = −∂p∂y,
which means that the pressure depends on x only. The firstmomentum equation (6) then becomes
0 = −dpdx
+ µd2u
dy2 . (9)
The partial derivatives are here replaced with ordinary deriva-
tives because p = p(x) and u = u(y). Moreover, sincedpdx
does
not depend on y and µd2udy2 does not depend on x, (9) can only be
satisfied if both these quantities are constants. Then the generalsolution of (9) is
p = p′x+ const, u =p′
2µy2 + C1y + C2, (10)
where p′ is a constant pressure gradient and C1 and C2 are ar-bitrary constants. Selecting different values of these parametersgenerates different solutions. Note that the constant enteringthe pressure is of no significance, as it neither enters the equa-tions nor typical boundary conditions.
We will consider two particular solutions.
6.1 Couette Flow 33
6.1 Couette Flow
Let’s consider a two-dimensional incompressible plane (∂/∂z =0) viscous flow between two parallel plates a distance 2h apart,figure 12. An important assumption we are making here is thatthe plates are very wide (into the page) and very long (upstreamand downstream of the section shown). The upper plate is madeto move at constant velocity V but there is no pressure gradientin the fluid, that is p′ = 0.
Figure 12: Incompressible viscous flow between parallel plates with no pres-sure gradient and with the upper plate moving.
The section shown is assumed to be far from the entrance tothe channel and hence is said to be fully developed and free ofentrance effects.
Since p′ = 0, the solution is u = C1y + C2 and the constantsare found from the boundary conditions: at y = +h u = V ;at y = −h u = 0 giving C1 = V
2h and C2 = V2 . Therefore the
particular solution is:
u =V
2hy +
V
2, (11)
where −h ≤ y ≤ h. This is called a Couette flow due to amoving wall with no slip at each wall.
6.2 Plane Poiseuille Flow
Let’s now consider the two plates to be fixed but that the pres-sure varies along the x coordinate. Then p′ = const 6= 0 in(10).
34 6 UNIDIRECTIONAL STEADY FLOWS
Figure 13: Incompressible viscous flow between parallel plates with a pressuregradient.
Using the boundary conditions u = 0 at y = ±h leads to theparticular solution:
u = − p′
2µ
(1− y2
h2
)(12)
Hence, in a Poiseuille flow the velocity profile is a parabolawith zero velocity at the walls and maximum velocity at thecentre-line.
35
7 Dynamic similarity of fluid flows
This is the most important section of the course. Pay attention,and study thoroughly!
7.1 Non-dimensional Navier-Stokes equations
Two geometric shapes are called similar if they can be madeidentical by scaling. Two flows are also called similar if they canbe made identical by scaling. In case of fluid flows scaling canbe applied both to the coordinates and to the flow parameterslike velocity and pressure. Consider two flows past two bodies ofgeometrically similar shapes, say, airfoils A and B, described bytwo velocity and pressure fields, say, uA(tA,xA), pA(tA,xA) anduB(tB,xB), pB(tB,xB). Fluid density and kinematic viscosityare ρA, νA and ρB, νB respectively. Both flows satisfy the Navier-Stokes equations. They also satisfy the no-slip condition on therigid walls, that is the requirement that the velocity is equal tozero (we work in the frame of reference where the airfoils are notmoving: instead, the air is approaching them from upstream).Another boundary condition is imposed at infinity: uA(xA) →uA as xA → ∞. Without loss of generality we can assume thatthe uA is strictly horizontal, that is uA = (UA, 0, 0). Similarcondition is imposed on the other flow, too.
Let as now scale the flows. First, we scale the coordinatesso as to make the airfoils identical. This can be achieved if wetake xA = LAx and xB = LBx and choose LA and LB to beequal to the chords of the respective airfoils. We also requirethat both airfoils have the same angle of attack. Second, wescale the velocities with their values at infinity: uA = UAu anduB = UBu. This ensures that thus defined u will have the samemagnitude at infinity in both cases. Third, we scale the pressureand time as pA = ρAU
2Ap, tA = LA
UAt and pB = ρBU
2Bp, tB =
LB
UBt. Substituting these new variables into the Navier-Stokes and
continuity equations and the boundary conditions reduces themto the following non-dimensional form
36 7 DYNAMIC SIMILARITY OF FLUID FLOWS
∂ui∂t
+ uj∂ui∂xj
= − ∂p∂xi
+1
Re
∂2ui∂xj∂xj
, i = 1, 2, 3,
∂uk∂xk
= 0,
ui|body = 0, i = 1, 2, 3,
u1 → 1, u2 → 0, u3 → 0 as x→∞
(13)
where the Reynolds number Re = ReA = UALA/UA for the firstflow and Re = ReB = UBLB/UB. It is now clear that flows Aand B will be dynamically similar when ReA = ReB. (The sameresult can be obtained from dimensional analysis using the Πtheorem.)
7.2 Why similarity is important
Dynamic similarity has several uses.
• If the velocity and length scales used are relevant, the na-ture of the flow can be judged by the Reynolds numberalone rather than by a combination of µ, ρ, U∞ and L.
• If the velocity and length scales used are relevant, the Reynoldsnumber has physical meaning of the ratio of inertial forcesto viscous forces. Indeed, a rough estimate of the inertialforces is the dynamic pressure ρU 2
∞, while the rough esti-mate for the friction stress µ∂u/∂y can be taken as µU∞/L.Therefore, Re may be considered as a rough estimate of theratio of the inertial forces to viscous forces in the bulk ofthe flow1. Another way of understanding this is to noticethat the viscous term in (13) has a factor of 1/Re.
• Solving the non-dimensional system (13) for one Re pro-duces the solution for an infinite number of real flows pastgeometrically similar bodies all having the same Re.
1Note and caution: as the boundary layer theory will show, this estimate can be farfrom accurate in certain thin regions.
7.2 Why similarity is important 37
• Wind-tunnel experiment provides accurate information onthe real flow with the same Re.
Of these four uses, the first one is by far the most important.
38 8 VORTICITY
8 Vorticity
Figure 14: Vorticity
Vorticity is one of the most important and most useful notionsin fluid dynamics.
8.1 Definition and properties
Vorticity is the angular velocity of fluid particles multiplied by2. It is easier to explain it first for two-dimensional flows. Letus mark fluid particles along a segment of a line and followthe motion of these particles. Such a line is called a materialline. Our material line will be translated, rotated, stretchedand bent. If the segment length is very small, bending can beneglected, and we come to an idea of the angular velocity, thatis the speed of rotation of such a small segment. In a solidbody this would be enough. Fluid, however, can also undergodeformation. As a result, two intersecting material lines mighthave different angular velocities, as it is illustrated in the figure.
Then, what can be called an angular velocity of the fluid atthe point of intersection of these two material lines?
The answer is that one should only take mutually-perpendicularmaterial lines. Then it can be proved mathematically that thesum of the angular velocities of the lines does not depend on theorientation of those lines. This sum is called vorticity.
Note that if the angular velocities of these two segments aredifferent then they can become perpendicular only for an instant,and only at that instant the sum of the angular velocities equalsvorticity.
In the general three-dimensional case angular velocity is avector. A solid body rotation has an axis, the direction of which
8.2 Vorticity equation 39
is the direction of the vector of angular velocity, while the speedof rotation is the magnitude of this vector. Let us now drawthree mutually perpendicular segments of material lines throughthe same point in the fluid. Then the sum of the angular veloc-ities at which any two of these lines rotate around the third lineis the vorticity component along that third line.
Note that material lines can be stretched by the flow. Thisis important, but does not change the definition of vorticity.
If the velocity field u(x) is known then it is possible to cal-culate the angular velocity of any material line and, hence, thevorticity. The corresponding operator, which, applied to thevelocity field gives the vorticity field has three names and nota-tions: rot, curl, and ∇ × . Vorticity is usually denoted ω, and,hence
ω = rot u = curl u = ∇× u.
In Cartesian coordinates operator ∇ can be represented as a
vector of spatial derivatives: ∇ =(∂∂x, ∂∂y, ∂∂z
), and then∇×u
is a vector product of two vectors.If the vorticity distribution is known then it is possible to
calculate the corresponding velocity distribution. This will bediscussed later in the course.
Note that even when the vorticity is everywhere zero a fluidmotion is possible. Flow with zero vorticity are called irrota-tional or potential. There are effective mathematical tools forstudying such flows, and they too yield to simple intuitive in-terpretation.
8.2 Vorticity equation
The equation for ω can be obtained by applying ∇× to (5). Ithas the form
∂ω
∂t+ u · ∇ω = ω · ∇u + ν∇2ω. (14)
The pressure term disappears from this equation. Instead, wehave ω · ∇u. The effect of this term is easy to understand atan intuitive level. In standard textbooks on fluid dynamics it is
40 8 VORTICITY
demonstrated that as a result of this term the vorticity vectorbehaves as an infinitesimal element of a material line: it is notonly advected but also rotated and stretched. This is the mainreason why vorticity is so important in fluid dynamics. Advec-tion, rotation, and stretching of vorticity are all effects yieldingto an intuitive understanding, unlike the evolution of the pres-sure. Thinking in terms of vorticity and velocity is thus ofteneasier than thinking in terms of velocity and pressure. We willuse this in our brief look at the boundary layers.
In case of a two-dimensional flow the vorticity ω has only onecomponent ω perpendicular to the plane of the flow. From (14)in 2D case we have
∂ω
∂t+ u · ∇ω = ν∇2ω. (15)
In 2D case a steam function ψ can be introduced such that
u1 =∂ψ∂x2
, u2 = − ∂ψ∂x1
. In steady flows fluid particles follow
the streamlines ψ = const. Combining the definitions of ψ andω gives a simple equation ∇2ψ = −ω, which can help to analysethe velocity distribution for a given vorticity distribution.
8.3 Activity
In Flow Illustrator (http://www.flowillustrator.com/) large pos-itive and large negative values of vorticity are marked withcolour. Watch (from the library or make your own) the videosand observe the relationship between vorticity and the snowflakemotion. This will help to understand the relationship betweenthe velocity and vorticity fields in 2-dimensional flows. In par-ticular, notice that for large Reynolds numbers, say, above 1000,the coloured areas often move with fluid particles, while the fluidparticles rotate around spots of colour.
Read a section on bathtub vortex on the Flow Illustrator web-site (http://www.flowillustrator.com/bathtub-vortex.php) to un-derstand better the vorticity dynamics.
Since the air (and water) viscosity is small, in many real flowsthe Reynolds number is very large. It may seem that one shouldexpect viscous friction force to be Re times less than the pressureforce. However, even the early experiments showed that theactual friction force is much greater than this estimate. Theexplanation was found by Ludwig Prandtl at the beginning ofthe 20th century.
Consider the behaviour of the vorticity in the flow past anairfoil at high Re. Far upstream, the flow is uniform and, hence,vorticity is zero. If we neglect viscous diffusion as compared tothe advection then this zero vorticity will be advected into theentire flow domain. As the result, we can expect the flow tobe potential. However, a potential flow cannot satisfy both theimpermeability (normal velocity un = 0) and the no-slip (tan-gential velocity uτ = 0) conditions. The impermeability condi-tion, corresponding to the mass conservation, takes precedence,and the potential flow satisfies un = 0. As for the no-slip con-dition, viscosity effects compensate by diffusing vorticity fromthe body surface across the streamlines, so that there is a layer,shown with dashed line in figure 15, into which vorticity pen-etrates and makes the flow in this layer rotational (that is not
42 9 HIGH REYNOLDS NUMBER FLOWS
potential). This layer is called the boundary layer. When Reincreases the rate of viscous diffusion of vorticity decreases ascompared to the rate with which it is advected (swept down-stream) by the flow. As a result, the boundary layer thicknessdecreases as Re increases. In the boundary layer the velocitychanges from 0 at the wall to the velocity of the potential flowat the edge of the boundary layer. Since the boundary layer isthin, the velocity gradient inside it is greater than in the bulkof the flow. Hence, in agreement with (4), the friction is greaterthere.
Boundary layers are usually found in one of two states: lami-nar or turbulent. Fortunately, even if turbulent, boundary layersremain thin, and the flow outside of them remains potential, asfar as it is not separated. Deeper analysis of boundary layerswill be a part of other MSc modules.
The above explanation of the boundary layer phenomenon isan example of how vorticity-velocity formulation can be usedto understand the fluid flow avoiding the difficulty of determin-ing the pressure discussed in section 5. The boundary layertheory itself is another method of avoiding this difficulty. Thisis because inviscid, and in particular, potential flows are muchsimpler than viscous flows, and in such flows pressure can easilybe found. In the boundary layer the pressure is close to thepressure given by the potential flow equations. This approachworks very well, but it applies only to attached flows at highReynolds numbers.
9.2 Activity
Watch a formation of boundary layers in Flow Illustrator movies.
43
10 Potential flows
10.1 Complex potential
Since in absence of separation the high-Reynolds-number flowsoutside boundary layers are often potential, it is useful to un-derstand the properties of potential flows. Fortunately, in thecase of two-dimensional flows there is a remarkably simple wayof describing potential flows. Rigorous proofs of the statementsmade in this subsection will be given later in the MSc course.These statement, however, are so simple that they can be easilymemorized.
Two-dimensional flows have two velocity components, whichwe will denote u and v, and they depend on two coordinates,which we will denote x and y, with u being the velocity com-ponent in x-direction and v being the velocity component in y-direction. Then one can introduce a complex variable z = x+ iy
(where i =√−1), so that x is the real part of z and y is the
imaginary part of z. Remarkably, it turns out that if we nowtake any analytic function F (z) it will describe some potentialflow. The rigorous definition of an analytic function can befound in mathematical courses, but for many practical purposesit is sufficient to say that analytic functions are functions hav-ing derivatives with respect to z. Thus, F = z2, F = z/(z + 1)and F = sin z are analytic functions, while F = x − iy is not,even though one can find the value of F for any given value ofz, and even though one can calculate the partial derivatives ofthis function with respect to x and y.
Any analytic function F (z) can be considered as the com-plex potential of a certain potential flow. This correspondenceconsists in the following. One can distinguish the real and imag-inary parts of F :
F (z) = φ(x, y) + iψ(x, y) (16)
Then the real-valued function ψ(x, y) is the stream function ofthat flow, and the real-valued function φ(x, y) is the potentialof that flow. Moreover, in other parts of the course it will be
44 10 POTENTIAL FLOWS
shown that the real and imaginary parts of the derivative dF/dzprovide the velocity components of this flow:
dF
dz= u− iv. (17)
Note the minus sign in front of iv: this is not a misprint. Proofsaside, one can now write down any expression for F in termsof z, calculate dF/dz, separate the real and imaginary part andget the velocity field of some potential flow. Before doing this,one should consider a very useful notion of a stream function.
10.2 The streamfunction ψ
We have seen that in 2D incompressible flow, the continuityequation is given by
∂u
∂x+∂v
∂y= 0. (18)
If we take any function ψ(x, y) and define u and v as
u =∂ψ
∂y, v = −∂ψ
∂x(19)
then these u and v will satisfy (18). (Prove this.) Moreover, forany u and v satisfying (18) there is a corresponding function ψ
such that these u and v are given by (19). (Try to prove this,too.) The function ψ is called the streamfunction.
Recall now our definition of streamline: a curve in the flowfield which is everywhere tangential to the fluid velocity. Theslope of the streamline at a point in a 2D flow field having ve-locity components (u, v) is given by
dy
dx=v
u, (20)
an equation which mathematically defines the streamline. If wesubstitute (19) into (20) we obtain:
dy
dx=v
u=−∂ψ∂x∂ψ∂y
=⇒ ∂ψ
∂y· dy +
∂ψ
∂x· dx = 0, (21)
10.3 The velocity potential φ 45
which implies that the difference ψ2 − ψ1 of two values of ψ atany two points 1 and 2 on the same streamline equals zero andtherefore ψ is constant along a streamline. Therefore if we get toknow the function ψ(x, y), we could then plot lines of constantψ to obtain a family of streamlines. It is important to note thata streamline shares a condition with a solid boundary: there isno flow through the streamline or the boundary. It is also worthnoting that the difference in the streamfunction value ψ betweentwo streamlines is the volumetric flow between them.
Stream function can be introduced for any two-dimensionalincompressible flow. If the flow we are considering is also irro-tational (which is the same as potential), i.e.
ω =∂v
∂x− ∂u
∂y= 0, (22)
substituting (19) gives
∂2ψ
∂x2+∂2ψ
∂y2= 0, (23)
which is called Laplace’s equation.
In polar coordinates, the expressions for Vr and Vθ are:
Vr =1
r
∂ψ
∂θVθ = −∂ψ
∂r, (24)
(the reader is encouraged to confirm that these also satisfy thecontinuity equation).
10.3 The velocity potential φ
If the flow is irrotational, i.e.
ω =∂v
∂x− ∂u
∂y= 0, (25)
we can use a function φ(x, y) called the velocity potential where:
u =∂φ
∂x, v =
∂φ
∂y, (26)
46 10 POTENTIAL FLOWS
i.e. a velocity vector u = (u, v) can be written as:
u = (u, v) =
(∂φ
∂x,∂φ
∂y
)= ∇φ. (27)
If we now consider the continuity equation (18) and substi-tuting (26) into it we obtain
∂u
∂x+∂v
∂y=∂2φ
∂x2+∂2φ
∂y2= 0, (28)
which is again Laplace’s equation.
It can be shown that equi-potential lines (that is lines ψ(x, y) =const) are perpendicular to streamlines as shown in figure 16.
Figure 16: An aerofoil with streamlines and equi-potential lines.
In polar coordinates, again using the expressions for Vr andVθ (24), we obtain Laplace’s equation as:
∂2φ
∂r2+
1
r
∂φ
∂r+
1
r2
∂2φ
∂θ2= 0. (29)
10.4 Standard solutions
10.4.1 Uniform stream at U∞ at an angle of incidence α
The complex potential for this flow is:
F (z) = U∞ze−iα where z = x+ iy (30)
yielding
10.4 Standard solutions 47
Figure 17: A uniform stream at U∞ at an angle of incidence α.
F (z) = φ+ iψ = [U∞x cosα + U∞y sinα]
+i [U∞y cosα− U∞x sinα](31)
i.e.
φ = U∞x cosα+U∞y sinα ψ = U∞y cosα−U∞x sinα (32)
We could deduce the velocity field from
i. F (z) knowing that dFdz = u− iv,
ii) from the velocity potential φ given u = ∂φ∂x , v = ∂φ
∂y
iii) or from the streamfunction ψ given u = ∂ψ∂y , v = −∂ψ
∂x .
These yield
u = U∞ · cos(α), v = U∞ · sin(α). (33)
10.4.2 Source or sink at the origin
The complex potential for a source or sink is given by
F (z) =m
2πln z, (34)
wherem is the volume flow rate per unit depth and has a positivevalue for a source and a negative value for a sink. The readershould confirm that
48 10 POTENTIAL FLOWS
Figure 18: A source at the origin.
φ =m
2πln r ψ =
m
2πθ. (35)
The reader should also confirm that the velocity componentsare:
Vr =m
2πr, Vθ = 0. (36)
If the source/sink is not at the origin but at z = z1, then
F (z) =m
2πln (z − z1). (37)
10.4.3 Point vortex
A point vortex is one where we have only a Vθ velocity compo-nent which varies as 1/r. The flow is irrotational everywhereexcept at the vortex axis (r = 0). Its complex potential is givenby
F (z) =−iΓ2π
ln z. (38)
The reader should confirm that
φ =Γ
2πθ, ψ =
−Γ
2πln r. (39)
10.5 Superposition of solutions 49
Figure 19: Irrotational vortex
The reader should also confirm that the velocity componentsare:
Vθ =Γ
2πr, Vr = 0. (40)
If the point vortex is not at the origin but at z = z1, then
F (z) =−iΓ2π
ln (z − z1). (41)
10.5 Superposition of solutions
The equations for incompressible, steady, irrotational flow arelinear and this fundamentally implies that if two solutions to anequation are known, then they could be added together to obtaina third solution. Therefore we could for example use the threestandard solutions developed in the preceding section to formmore complex flow situations. We could write an expression forthe complex function F (z) (as long as the function is analytic)and then we explore what is the resulting flow field.
10.5.1 Flow due to a combined source and uniform stream
We could for example combine a uniform stream at zero inci-dence and a source (for example located at the origin). The
50 10 POTENTIAL FLOWS
complex potential, the streamfunction and the velocity poten-tial are simply the sums of the individual quantities:
Figure 20: Combined source in a uniform stream
F (z) = U∞z +m
2πln z, (42)
φ = xU∞ +m
2πln r, (43)
ψ = yU∞ +m
2πθ. (44)
Figure 21: Streamlines due to a source in a uniform stream.
10.5.2 Flow due to a combined source, sink and uniform stream
If we combine a source at z = −z1, a sink at z = z1 and auniform stream, we obtain something usually referred to as a
10.5 Superposition of solutions 51
Rankine oval, a rather streamlined ‘body’ in a flow. The readeris encouraged to construct the streamfunction (simply addingthe streamfunction relationships of a source, a sink and a uni-form stream) and plotting the result.
10.5.3 Flow around a circular cylinder
On the other hand, flow around a circular cylinder has the fol-lowing complex potential, streamfunction and velocity potential(stated here without proof or derivation):
F (z) = U∞
(z +
R2
z
)− iΓ
2πln z, (45)
φ = U∞ cos θ
(r +
R2
r
)+
Γ
2πθ, (46)
ψ = U∞ sin θ
(r − R2
r
)− Γ
2πln r, (47)
where r and θ are polar coordinates with origin at the cylin-der centre. At radius r = R the radial velocity is zero and thestreamline is circular forming the virtual cylinder ‘solid’ body.For r > R we can then observe the flow around the circularcylinder of radius R. Note that due to the Γ parameter the so-lution is non-unique but depends on the circulation. Refer to thecorresponding sections of http://www.flowillustrator.com for ashort discussion and an animation. The reader is encouragedto code and plot the streamlines and the equi-potential lines intheir favourite plotting program.
10.5.4 Flow around a corner of arbitrary angle
Flows around corners (stated here without proof or derivation)take the relatively simple form of:
F (z) = zn = rneinθ, (48)
where n is a constant. This implies that:
φ = rn cosnθ, ψ = rn sinnθ. (49)
52 10 POTENTIAL FLOWS
Figure 22: Streamlines around a corner of 90 (π/2) degrees.
Figure 23: Streamlines around an internal corner of 60 (π/3) degrees.
Using the result of figure 23 and expanding the plot to dou-ble the angular extent, we reveal (see figure 25) that in contrastto the case of figure 24 where an erroneous infinite velocity oc-curred, in this case, a stagnation flow towards the corner occurs.
10.6 Exercise 53
Figure 24: Streamlines around an external corner of 90 (π/2) degrees. Thevelocity at the corner turns out to be infinite. The flow downstream of thecorner is also unrealistic, since in reality separation would occur.
Figure 25: Streamlines around an internal corner of 120 degrees. The partingstreamline reveals flow which goes to stagnation at the corner.
10.6 Exercise
For the case where a source at the origin is in a uniform stream,find the location of the stagnation point. Hint: you will needto evaluate the u and v components of velocity and then thinkwhat happens at a stagnation point. Answer: Stagnation pointlies m
2πU∞ahead of the origin (where the centre of the source is
located).
54 11 CONCLUSION
11 Conclusion
In this self-taught course we have met fundamental concepts influid mechanics striking a balance between the physical descrip-tion and the mathematical background. You will revisit severalof the concepts presented here in other taught subjects duringthe MSc course but in the process more advanced material willbe presented.
55
A Appendix: quick review of vector nota-
tion.
In a Cartesian coordinate system (figure A) a vector velocityu is given by u = ui + vj + wk, where i, j and k are mutu-ally perpendicular vectors of unit length, and u, v and w arecomponents of velocity along each direction respectively.
Figure 26: Cartesian coordinate system
The vector u can also be written as a column vector
u =
u
vw
.
Another way to write down a vector via its components in-volves summation over repeated indexes. First, we introduce analternative notation for the unit vectors i = e1, j = e2, k = e3,and for the components: u = u1, v = u2, w = u3. Then
u =n=3∑n=1
unen = unen.
In the last term the summation sign is omitted, but the summa-tion is still implied. This is a widely used and very convenientconvention. It was introduced by A. Einstein, who used to saythat it was the greatest of his discoveries. Importantly, the sum-mation is implied only over repeated indices. For example, anbnalways denotes a1b1+a2b2+a3b3, but anbm is simply a product ofan and bm without any summation. Note also that the repeatedindex is mute in the sense that aibi = akbk, that is the letter
56 A APPENDIX: QUICK REVIEW OF VECTOR NOTATION.
used to denote the repeated index does not affect the mean-ing: indeed, if the expression with a repeated index is unfoldedthis letter disappears anyway. In the following formulae we willgive two expressions, one via u, v, and w and another using thesummation over repeated indices.
Note that the expressions for various operators via the com-ponents of the vector given below are valid only in Cartesian co-ordinate systems. In, say, a cylindrical coordinate system theywill be different. Expressions for other coordinates systems canbe obtained by substitutions.
The dot product of u with itself is a scalar, which is obtainedby squaring the individual elements and adding
u · u = u2 + v2 + w2 = unun.
In what follows we consider vector fields, which means that uvaries from point to point in space. We denote x = x1, y = x2,
and z = x3 the coordinates in this space.The divergence of the vector u is a scalar quantity given by
∇ · u =∂u
∂x+∂v
∂y+∂w
∂z=∂un∂xn
.
The gradient of a scalar φ is a vector, given by
∇φ =
∂φ∂x∂φ
∂y∂φ∂z
=∂φ
∂xnen.
The curl of the velocity vector is a vector (called the vortic-ity), given by a determinant
∇× u =
∣∣∣∣∣∣∣i j k∂∂x
∂∂y
∂∂z
u v w
∣∣∣∣∣∣∣ =
∂w∂y− ∂v∂z
∂u∂z− ∂w∂x
∂v∂x− ∂u∂y
(we do not use implicit summation here).
57
The Laplacian of a scalar φ is a scalar, given by
∇2φ =∂2φ
∂x2 +∂2φ
∂y2 +∂2φ
∂z2 =∂2φ
∂xi∂xi.
Note how the summation convention is used to denote secondderivatives in this expression. Laplacian has also a special no-tation ∆, so that ∆φ and ∇2φ are the same. However, ∆ is alsooften used to denote not the Laplacian but the increment of aquantity.
The Laplacian of a vector u is a vector, given by
∇2u =
∂2u∂x2 + ∂2u
∂y2 + ∂2u∂z2
∂2v∂x2 + ∂2v
∂y2 + ∂2v∂z2
∂2w∂x2 + ∂2w
∂y2 + ∂2w∂z2
=∂2u
∂xi∂xi=
∂2un∂xi∂xi
en.
Here are some useful vector identities (see Maths texts forthe proofs):
