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

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1 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 streamline Streamlines cannot intersect each other Velocity vector of each article occying a oint on the streamline is tangential to the streamline. Streamtbe A bndle of streamlines bonded on the otside by streamlines. Flid cannot cross the streamtbe bondary. Only enters and leaves via the tbe ends. Pathline The ath taken by a flid article as it is translated in sace. Streakline An instantaneos line whose oints are occied by all articles originating from some secified oint in the flow.

2 Tyes of Flow Uniform Non-niform Steady Flow at a high constant rate throgh a long straight ie of constant cross-section Flow at constant rate throgh a taering ie Unsteady High velocity flow accelerating or decelerating throgh long straight ie of constant cross-section Accelerating or decelerating flow throgh a taering ie Note we need to define or reference axes, for the flow is described as steady with reference to where we are viewing the flow. Consider, for examle, two ersons viewing the flow of water in a river, on one the bank, one in a boat moving at the velocity of the river flow: Passenger in the bow viewing the flow will see a flow attern stationary in time. Person on shore will see flow changing ast the reference oint. Viscos and Inviscid Flow An ideal flid is defined as an incomressible flid having negligible viscosity. Inviscid flow: Viscos effects do not significantly inflence the flow and are ths neglected. Examles are external flows flows which exist exterior to a body. An examle is flow arond an airfoil or a hydrofoil. Viscos flow: Viscos effects cannot be ignored. Examles are flow in ies and oen channels. Continity Eqation Princile of Conservation of Mass Consider fixed region within the flid:

3 Rate at which mass enters the region rate at which mass Rate of accmlation of + () leaves the region mass in the region Since the flow is steady, then the rate at which mass accmlates 0 Consider a streamtbe of small cross-sectional area δa sch that velocity is constant over it. Now δs is so small that δa B δa C δa Therefore, rate of flid volme assing throgh lane C from lane B is δs ds δ A δa as t 0 () δt dt Bt ds dt linear velocity, (3) Therefore, the rate at which mass of flid enters a selected ortion of a streamtbe having area δa, velocity,, and density ρ is ρ δa If flow is steady, then at another cross-section () with δa, velocity,, and density ρ ρ δa ρ δa LL constant (4) Consider the entire collection of streamtbes, ρ da cons tant (5) For ρ and constant, then, ρ A cons tant (6) For constant density da cons tant (7) A cons tan t (8) Note that from this, it imlies that as A increases, the velocity decreases. 3

4 Acceleration of a flid article a. Along direction of flow Consider a article in a stream of flid, initially at oint A, bt then moving to oint B. Its velocity changes by virte of change in Its osition Time Assme that as it moves δs from A to B in time δt reslting in a change of its velocity by δ. δ δ s + δ t (9) s As δ t 0 d ds a s + (0) dt s dt Bt ds () dt a s + () s a s is the sbstantial acceleration; the first term on the RHS is the convective acceleration; and the second term on the RHS is the local acceleration For steady flow, 0 Therefore, a s (3) s b. Acceleration Perendiclar to flow Particle moving in crved ath changes direction. This imlies that comonent of acceleration towards the center of crvatre of the ath. 4

5 Let radis of crvatre of streamline be r s Therefore, acceleration towards the centre (4) r s Temoral acceleration n (5) Where n is velocity comonent toward centre of crvatre. So an r s + n (6) Bernoilli s Eqation 5

6 In the absence of shearing forces, any force acting on a srface is erendiclar to it. Assmtions: Inviscid flow no viscosity; steady flow; only forces de to gravity and ressre are significant. Consider the figre above. Assme average vale of the ressre at the sides is +kδ (k<.0) The resltant force on streamtbe in the direction of flow is therefore given by: A ( + δ)( A + δa) + ( + kδ) δa ρgaδs cosθ Ignoring second order of small qantities, above becomes: Aδ ρgaδs cosθ This net force, by Newton s nd Law eqals rodct of mass and acceleration in the direction of the force That is: d ρ Aδ s dt Aδ ρgaδ s cosθ (7) Now δ s cos θ δ z (8) Dividing by ρ Aδ s and taking the limit as ρδ s 0 d d dz + + g 0 ρ ds dt ds (9) Now d + dt s (0) as 0 s (steady flow) () For constant density flid, eqation can be integrated to give 6

7 + ρ + gz constant () or ρg + g + z constant (3) Known as Bernoilli s eqation Holds for:. steady flow. flow along a streamline 3. inviscid flow 4. constant density Bernoilli s Eqation exresses the transformation from one form of energy into another. Energy Transformation in a Constant-density Flid HEAD ENERGY PER UNIT WEIGHT 7

8 Total Head Line: z + + ρ g g Pressre line/hydralic grade line Static ressre head line ρg + + z constant g ρg z + ρg (3) At P, 0 g Hence z + constant h + z H ρg Note elevations in the iezometers at () and (3). Alications of Bernoilli s Eqation Energy Transformation in a Constant Density Flid See diagram in Doglas, 6.5 (g 80) Energy at () Energy at () Energy at (3) Bernoilli s Eqation Head Energy er nit weight At () + + ρ gz g constant (3) Orifice Refer to the figre below 8

9 () and () are on the same streamline Shar edge orifice imlies minimal contact with flid, which means redced friction losses. Large reservoir imlies that the velocity at srface is aroximately 0. Liqid as free jet imlies nder the inflence of gravity. Flow convergence occrs at a lace called the vena contracta. Pressre at the vena contracta is niform and eqal to atmosheric ressre, P a For steady flow and low friction, the Bernoilli s eqation can be alied. Therefore, ρ g + g + z Patm ρ g + g + 0 (33) Assme 0 (very large reservoir and oint removed from ()) Therefore, ρ g h (34) or ( ) z ρ g h + z (35) Therefore, h (36) g 9

10 gh (37) For real flids (with friction) Actal (mean) velocity C v gh (C v is the coefficient of velocity) Then, the actal area at vena contracta < A orifice So a coefficient of contraction C c is introdced Coefficient of discharge: Actal discharge xv vc actal C d Ideal discharge Aorifice xvideal A C C c v (38) Pitot Tbe Refer to Doglas (Section 6.6, 6.7) for treatment of the Pitot tbe Note discssion of the Stagnation Point oint in a flid stream where the velocity is redced to zero Ventri-Meter Refer to Doglas (6.0) and the lab notes for treatment of the Ventri-Meter 0

STATIC, STAGNATION, AND DYNAMIC PRESSURES

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