Chapter 3 Bernoulli Equation3.1 Flow Patterns: Streamlines, Streaklines, Pathlines3.2 Streamline Coordinates3.3 Euler Equation3.4 Bernoulli Equation1) Along a streamline2) Irrotational flow3) Unsteady irrotational flow4) Streamline coordinatesAlong a streamlineNormal to a streamline3.5 Physical interpretation of Bernoulli equation3.6 Static, Stagnation, Dynamic, and Total Pressure3.7 Applications of Bernoulli Equation1) Stagnation Tube2) Pitot Tube3) Free Jets4) Simplified form of the continuity equation5) Volume Rate of Flow (flowrate, discharge)1. Cross-sectional area oriented normal to velocity vector2. General case6) Flowrate measurement3.8 Energy grade line (EGL) and hydraulic grade line (HGL)3.9 Limitations of Bernoulli Equation1) Compressibility Effects:2) Unsteady Effects:3) Rotational Effects4) Other RestrictionsChapter 3157:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2007Chapter 3 Bernoulli Equation3.1 Flow Patterns: Streamlines, Streaklines, PathlinesA streamline is a line that is everywhere tangent to the velocity field.Streamlines are obtained analytically by integrating theequations defining lines tangent to the velocity field. Fortwo-dimensional flows the slope of the streamline, dy /dx,must be equal to the tangent of the angle that the velocityvector makes with the x axis ordydx=vuIf the velocity field is known as a function of x and y(and t if the flow is unsteady), this equation can be integrated to give the equa-tion of the streamlines.A pathline is the line traced out by a given particle as it flows from onepoint to another. The pathline is a Lagrangian concept that can be produced inthe laboratory by marking a fluid particle (dying a small fluid element) and takinga time exposure photograph of its motion.Illustration of a pathline (left) and an example of pathlines, motion of water induced by sur -face waves (right).Chapter 3257:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2007A streakline consists of all particles in a flow that have previously passedthrough a common point. Streaklines are more of a laboratory tool than an ana-lytical tool. They can be obtained by taking instantaneous photographs of markedparticles that all passed through a given location in the flow field at some earliertime. Such a line can be produced by continuously injecting marked fluid (neu-trally buoyant smoke in air, or dye in water) at a given location. Illustration of a streakline (left) and an example of streaklines, flow past a full-sized stream-lined vehicle in the GM aerodynamics laboratory wind tunnel, and 18-ft by 34-ft test sectionfacility driven by a 4000-hp, 43-ft-diameter fan (right).If the flow is steady pathlines, streamlines, and streaklines are the same.For unsteady flows none of these three types of lines need be the same. 3.2 Streamline CoordinatesIn the streamline coordinate system the flow is described in terms of onecoordinate along the streamlines, denoted s, and the second coordinate nor-mal to the streamlines, denoted n. Unit vectors in these two directions are de-noted by ^s and ^n, as shown in Fig. 4.8. Of the major advantages of using thestreamline coordinate system is that the velocity is always tangent to the s di-rection. That is,V =vs^sThis allows simplifications in describing the fluid particle acceleration and in solv-ing the equations governing the flow.Chapter 3357:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2007Figure 4.8 Streamline coordinate system for two-dimensional flow.For two-dimensional flow we can determine the acceleration asa=D VDt= as^s+an^nwhere as and an are the streamline and normal acceleration, respectively. Ingeneral, for steady flow both the speed and the flow direction are a function of lo-cation V =V(s, n) and ^s=^s(s, n). Thus, application of the chain rule givesa=D(vs^s)Dt=D vsDt^s+vsD^sDtora=(∂ vs∂t+∂ vs∂ sdsdt+∂ vs∂ ndndt)^s +vs(∂^s∂ t+∂^s∂ sdsdt+∂^s∂ ndndt)This can be simplified by using the fact that the velocity along the streamline is vs=dsdtand the particle remains on its streamline (n = constant) so that dndt=0 Hence,Chapter 3457:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2007a=(∂ vs∂t+vs∂ vs∂ s)^s+vs(∂^s∂ t+vs∂^s∂ s)Figure 4.9 Relationship between the unit vector along the streamline, ^s, and the radius ofcurvature of the streamline, RThe quantity ∂^s/∂ s represents the limit as δs →0 of the change in theunit vector along the streamline, δ^s, per change in distance along the stream-line, δs. The magnitude of ^s is constant (|^s|= 1; it is a unit vector), but itsdirection is variable if the streamlines are curved. δ^s=^s(s+δs)−^s(s)=(^s+∂^s∂ sδs)−^s=∂^s∂ sδsIn the limit δs →0, the direction of δ^s/δs is seen to be normal to the stream-line. That is,δ^s=|^s|δθ^n=δθ^nThus, by equating the above two equations of δ^s∂^s∂ sδs=δθ^n∂^s/∂ s can be expressed as∂^s∂ s=δθδs^n=δθR δθ^n=^nRChapter 3557:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2007where δs=R δθ from Fig. 4.9.Similarly, the quantity ∂^s/∂ t represents the limit as δt →0 of thechange in the unit vector along the streamline, δ^s, per change in time, δt.δ^s=^s(t +δt)−^s(t)=(^s+∂^s∂tδt)−(^s)=∂^s∂ tδtIn the limit δt →0, the direction of δ^s/δ t is also seen to be normal to thestreamline. That is,∂^s∂ tδt=δθ^nThus, as δt →0∂^s∂ t= limδt→ 0δθδt^n=∂ θ∂t^nFinally by multiplying vs with ∂^s/∂ t we getvs∂ θ∂ t^n=∂ vn∂t^nNote that the unit vector ^n is directed from the streamline toward the center ofcurvature.Hence, the acceleration for two-dimensional flow can be written in terms ofits streamwise and normal components in the forma=(∂ vs∂t+vs∂ V∂ s)^s+(∂ vn∂ t+vs2R)^noras=∂ vs∂ t+vs∂ vs∂ s, an=∂ vn∂ t+vs2RThe first term ∂ vs/∂t represents the local acceleration in streamline direc-tion, the second term vs∂ vs/∂ s represents the convective acceleration along thestreamline due to the spatial gradient i.e., convergence/divergence streamlines,Chapter 3657:020 Mechanics of Fluids and Transport ProcessesProfessor Fred Stern Fall 2007the third term ∂ vn/∂ t represents the local acceleration normal to flow, and thelast term vs2/ R represents centrifugal
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