STATIC, STAGNATION, AND DYNAMIC PRESSURES

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1 STATIC, STAGNATION, AND DYNAMIC PRESSURES Bernolli eqation is g constant In this eqation is called static ressre, becase it is the ressre that wold be measred by an instrment that is static with resect to the flid. Of corse, if the instrment were static with resect to flid, it wold have move along with the flid. However, sch a measrement rather difficlt to make in a ractical sitation. However, we showed that there was no ressre variation normal to straight streamlines. This fact makes it ossible to measre the static ressre in a flowing flid sing a wall ressre ta laced in a region where the flow streamlines are straight as shown in the figre. The ressre ta is a small hole, drilled careflly in the wall, with its axis erendiclar to the srface. Figre. Measrement of static ressre. MM3 5

2 In a flid stream far from a wall, or where streamlines are crved, accrate static ressre measrements can be made by carefl se of a static ressre robe, shown in the figre. When a flowing flid is decelerated to ero seed by a frictionless rocess, the ressre is measred at that oint is called stagnation ressre. Figre. Measrement of stagnation ressre (Pitot tbe). In incomressible flow, alying Bernolli eqation between oints in the free stream and at the nose of tbe and taking = at the tbe centerline, we get where P is the stagnation ressre, the stagnation seed is ero. where is the static ressre. The term generally is called dynamic ressre. Solving the dynamic ressre gives and for the seed MM3 5

3 The static ressre corresonds to a oint A is read from the wall static ressre ta. The stagnation ressre is measred directly at A by the total head tbe. Two robes are combined as in itot-static tbe. The inner tbe is sed to measre the stagnation ressre at oint B while the static ressre at C is measred by the small holes in the oter tbe. MM3 5 3

4 Examle: A simle itot tbe and a ieometer are installed in a vertical ie as shown in the figre. If the deflection in the mercry manometer is. m, then determine the velocity of the water at the center of the ie. The densities of water and mercry are kg/m 3 and 36 kg/m 3, resectively. MM3 5 4

5 RELATION BETWEEN THE FIRST LAW OF THERMODYNAMICS AND THE BERNOULLI EQUATION Consider steady flow in the absence of shear forces. We choose a control volme bonded by streamlines along its erihery. Sch a control volme often is called a streamtbe. We aly energy eqation to this control volme. Basic eqation (Energy eqation) Q Restrictions: ) W s ) W shear 3) W other 4) Steady flow 5) Uniform flow and roerties at each section Under these restrictions t W s Wshear Wother ed e C CS g da Bt from continity nder these restrictions d da t or A A C CS A g A Q MM3 5 5

6 That is, Also, Ths, from the energy eqation or Under the restriction of incomressible flow and hence This will redce to the Bernolli eqation if the term in arentheses were ero. Ths, nder the additional restrictions, 6) incomressible flow 7) The energy eqation redces to Before, the Bernolli eqation was derived from momentm considerations (Newton s second law), and is valid for steady, incomressible, frictionless flow along a streamline. In this section, the Bernolli eqation was obtained by alying the first law of thermodynamics to a streamtbe control volme, sbject to restrictions throgh 7 above. A A m m dm Q dt dm dm Q t Q Q m dm Q m g g dm Q g g dm Q g g constant dm Q constant g g MM3 5 6

7 Examle: Consider the frictionless, incomressible flow with heat transfer. Show that Q dm MM3 5 7

8 ENERGY GRADE LINE AND HYDRAULIC GRADE LINE Often it is convenient to reresent the mechanical energy level of a flow grahically. The energy eqation, that is Bernolli eqation, sggests sch a reresentation. Dividing Bernolli eqation by g, we obtain g g H constant Each term has dimensions of length, or head of flowing flid. The individal terms are g g H is the head de to local static ressre is the head de to local dynamic ressre is elevation head is the total head of the flow The energy grade line (EGL): The locs of oints at a vertical distance, H, measred above a horiontal datm, g g which is the total head of the flid. The hydralic grade line (HGL): The locs of oints at a vertical distance, g, measred above a horiontal datm. The difference is heights between the EGL and HGL reresents, the dynamic (velocity) head,. g MM3 5 8

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10 UNSTEADY BERNOULLI EQUATION INTEGRATION OF EULER S EQUATION ALONG A STREAMLINE Consider the streamwise Eler eqation in streamline coordinates The above eqation may now be integrated along an instantaneos streamline from oint to oint to yield For an incomressible flow, it becomes Restrictions: ) Incomressible flow ) Frictionless flow 3) Flow along a streamline t s g s s ds t ds s g ds s ds s ds t g g s MM3 5

11 Examle: A long ie is connected to a large reservoir that initially is filled with water to a deth of 3 m. The ie is 5 mm in diameter and 6 m long. As a first aroximation, friction may be neglected. Determine the flow velocity leaving the ie as a fnction of time after a ca is removed from its free end. The reservoir is large enogh so that the change in its level may be neglected. MM3 5

12 IRROTATIONAL FLOW When the flid elements moving in a flow field do not ndergo any rotation, then the flow is known to be irrotational. For an irrotational flow, that is, In cylindrical coordinates, BERNOULLI EQUATION APPLIED TO IRROTATIONAL FLOW Eler eqation for steady flow was sing vector identity We see that for irrotational flow ; therefore, it redces to And Eler s eqation for irrotational flow can be written as or w y x v x w v y w r r r r r r r g g MM3 5

13 Dring the interval dt, a flid article moves from the vector osition to the osition r dr. Taking the dot rodct of dr dxı dyj dk with each of the terms in above eqation, we obtain and hence dr g dr d gd d integrating this eqation gives, d g constant For incomressible flow, = constant, and g constant Since dr was an arbitrary dislacement, this eqation is valid between any two oints in the flow field. The restrictions are. Steady flow. Incomressible flow 3. Inviscid flow 4. Irrotational flow dr r MM3 5 3

14 ELOCITY POTENTIAL We can formlate a relation called the otential fnction,, for a velocity field that is irrotational. To do so, we mst se the fndamental vector identity crl grad which is valid if (x,y,,t) is a scalar fnction, having continos first and second derivatives. Then, for an irrotational flow in which, a scalar fnction,, mst exist sch that the gradient of is eqal to the velocity vector,. Ths, x v y w In cylindrical coordinates r r r The otential velocity,, exists only for irrotational flow. Irrotationality may be a valid assmtion for those regions of a flow in which viscos forces are negligible. For examle, sch a region exists otside the bondary layer in the flid over a solid srface. All real flids ossess viscosity, bt there are many sitations in which the assmtion of inviscid flow considerably simlifies the analysis and gives meaningfl reslts. MM3 5 4

15 STREAM FUNCTION AND ELOCITY POTENTIAL FOR TWO-DIMENSIONAL, IRROTATIONAL INCOMPRESSIBLE FLOW; LAPLACE S EQUATION For two dimensional, incomressible, inviscid flow, velocity comonents and v can be exressed in terms of stream fnction,, and the velocity otential,, y x v x v y Sbstitting for and v into the irrotational condition v weobtain x y x y (A) Sbstitting for and v into the continity eqation x weobtain v y x y (B) Eqations (A) and (B) are forms of Lalace s eqation. Any fnction or that satisfies Lalace s eqation reresents a ossible two dimensional, incomressible, irrotational flow field. MM3 5 5

16 Along a streamline, stream fnction is constant, therefore d dx dy x y The sloe of a streamline becomes dy dx / x / y v v Along a line of constant, d = and d dx dy x y Conseqently, the sloe of a otential line becomes dy dx / x / y v As otential lines and streamlines have sloes that are negative recirocals; they are erendiclar. MM3 5 6

17 Examle: Consider the flow field given by = 4x 4y. Show that the flow is irrotational. Determine the stream fnction for this flow. MM3 5 7

18 ELEMENTARY PLANE FLOWS A variety of otential flows can be constrcted by serosing elementary flow atterns. The and fnctions for five elementary two dimensional flows a niform flow, a sorce, a sink, a vortex and a doblet are smmaried in the Table below. MM3 5 8

19 SUPERPOSITION OF ELEMENTARY PLANE FLOWS We showed that both and satisfy Lalace s eqation for flow that is both incomressible and irrotational. Since Lalace s eqation is a linear homogeneos artial differential eqation, soltions may be serosed (added together) to develo more comlex and interesting atterns of flows. Table. Serosition of Elementary Plane Flows MM3 5 9

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22 Examle: A sorce with strength. m 3 /s m and a conterclockwise vortex with strength m 3 /s are laced on origin. Obtain stream fnction an velocity otential, and velocity field for the combined flow. Find the velocity at oint (,.5). MM3 5

23 Examle: The following stream fnction reresents the flow ast a cylinder of radis a with circlation. Ua UrSin r r Sin au ln a Determine the ressre distribtion over the cylinder. MM3 5 3

Fluid Dynamics. Type of Flows Continuity Equation Bernoulli Equation Steady Flow Energy Equation Applications of Bernoulli Equation

Fluid Dynamics. Type of Flows Continuity Equation Bernoulli Equation Steady Flow Energy Equation Applications of Bernoulli Equation Tye of Flows Continity Eqation Bernolli Eqation Steady Flow Energy Eqation Alications of Bernolli Eqation Flid Dynamics Streamlines Lines having the direction of the flid velocity Flids cannot cross a