Chapter 2: Pressure Distribution in a Fluid

Transcription

1 ME:56 Chapter Professor Fred Stern Fall 7 Chapter : Pressure Distribution in a Fluid Pressure and pressure gradient In fluid statics, as well as in fluid dnamics, the forces acting on a portion of fluid (CV) bounded b a CS are of two kinds: bod forces and surface forces. Bod Forces: act on the entire bod of the fluid (force per unit volume). Surface Forces: act at the CS and are due to the surrounding medium (force/unit areastress). In general the surface forces can be resolved into two components: one normal and one tangential to the surface. Considering a cubical fluid element, we see that the stress in a moving fluid comprises a nd order tensor. σ Face σz σ Direction z

2 ME:56 Chapter Professor Fred Stern Fall 7 σ ij σ σ σ z σ σ σ z σ z σ z σ zz Since b definition, a fluid cannot withstand a shear stress without moving (deformation), a stationar fluid must necessaril be completel free of shear stress (σij, i j). The onl non-zero stress is the normal stress, which is referred to as pressure: n σii-p σ n -p, which is compressive, as it should be since fluid cannot withstand tension. (Sign convention based on the fact that p> and in the direction of n) Or p p p z p n p (one value at a point, independent of direction; p is a scalar) i.e. normal stress (pressure) is isotropic. This can be easil seen b considering the equilibrium of a wedge shaped fluid element

3 ME:56 Chapter Professor Fred Stern Fall 7 3 F : p dasinα p dasinα n p p n F : p dacosα + p dacosα W z n z Where: W γv V z l cosα z l sin α l pnda dl da z α pdasinα α dadld WρgVγV pzdacosα da / dl W p n γda cosα dl sin α p n p da n + p z p z cosα + for dl p da z γ dl sin α i.e. cosα γda cosα dl sin α p n p p p z Note: For a fluid in motion, the normal stress is different on each face and not equal to p. σ σ σzz -p B convention p is defined as the average of the normal stresses p ( σ + σ + σzz ) σii 3 3

4 ME:56 Chapter Professor Fred Stern Fall 7 4 The fluid element eperiences a force on it as a result of the fluid pressure distribution if it varies spatiall. Consider the net force in the direction due to p(,t). d d dz The result will be similar for df and dfz; consequentl, we conclude: p ˆ p ˆ p df ˆ press i j k z Or: f p force per unit volume due to p(,t). Note: if pconstant, f.

5 ME:56 Chapter Professor Fred Stern Fall 7 5 Equilibrium of a fluid element Consider now a fluid element which is acted upon b both surface forces and a bod force due to gravit df ρ gd or ρ g (per unit volume) grav f grav Application of Newton s law ields: ρd a ( f ) d ρa f f + bod f surface per unit d m a F f bod g and g gkˆ f ρg kˆ bod ρ z g f f + f surface pressure viscous (includes f, since in general σij viscous pδij τij f p pressure + ) Viscous part V V V f µ + + µ V viscous z For ρ, μconstant, the viscous force will have this form (chapter 4). V t ρa p+ ρ g+ µ V with a + V V inertial pressure gradient gravit viscous

6 ME:56 Chapter Professor Fred Stern Fall 7 6 This is called the Navier-Stokes equation and will be discussed further in Chapter 4. Consider solving the N-S equation for p when a and V are known. p ρ g a + µ V B(, t ( ) ) This is simpl a first order PDE for p and can be solved readil. For the general case (V and p unknown), one must solve the NS and continuit equations, which is a formidable task since the NS equations are a sstem of nd order nonlinear PDEs. We now consider the following special cases: ) Hdrostatics ( V a ) V ) ) Rigid bod translation or rotation ( 3) Irrotational motion ( ( a) ( a) a vector identit V ) V if ρ constant V Euler equation Bernoulli equation also, V V if const ϕ & ρ. ϕ

7 ME:56 Chapter Professor Fred Stern Fall 7 7 Case () Hdrostatic Pressure Distribution p ρg ρgk z g p p i.e. and p ρ g z dp ρgdz or p p ρ gdz g ρ( z) dz r g g r constant near earth's surface r liquids ρ constant (for one liquid) p -ρgz + constant gases ρ ρ(p,t) which is known from the equation of state: p ρrt ρ p/rt dp p g R dz T (z) which can be integrated if T T(z) is known as it is for the atmosphere.

8 ME:56 Chapter Professor Fred Stern Fall 7 8 Manometr Manometers are devices that use liquid columns for measuring differences in pressure. A general procedure ma be followed in working all manometer problems:.) Start at one end (or a meniscus if the circuit is continuous) and write the pressure there in an appropriate unit or smbol if it is unknown..) Add to this the change in pressure (in the same unit) from one meniscus to the net (plus if the net meniscus is lower, minus if higher). 3.) Continue until the other end of the gage (or starting meniscus) is reached and equate the epression to the pressure at that point, known or unknown.

9 ME:56 Chapter Professor Fred Stern Fall 7 9 Hdrostatic forces on plane surfaces The force on a bod due to a pressure distribution is: F pn da A where for a plane surface n constant and we need onl consider F noting that its direction is alwas towards the surface: F p da. A Consider a plane surface AB entirel submerged in a liquid such that the plane of the surface intersects the freesurface with an angle α. The centroid of the surface is denoted (, ). F γ sinαa pa Where p is the pressure at the centroid.

10 ME:56 Chapter Professor Fred Stern Fall 7 To find the line of action of the force which we call the center of pressure (cp, cp) we equate the moment of the resultant force to that of the distributed force about an arbitrar ais. cpf df A Note: df γsinαda γ sinα da da I moment of o A A + I A Inertia about O O I moment of inertia WRT horizontal centroidal ais F pa γ sinαa cpγ sinαa γ sinα( A+ I) cp + I A and similarl for cp cp cp F df A I + A where I I product of I + A inertia Note that the coordinate sstem in the tet has its origin at the centroid and is related to the one just used b: tet and tet ( )

11 ME:56 Chapter Professor Fred Stern Fall 7 Hdrostatic Forces on Curved Surfaces z In general, F A pn da Horizontal Components: F F i p n i da da F A p da da projection of n da onto a plane perpendicular to direction da projection of n da onto a plane perpendicular to direction The horizontal component of force acting on a curved surface is equal to the force acting on a vertical projection of that surface including both magnitude and line of action and can be determined b the methods developed for plane surfaces. F pn k da p da γ h daz δ γ z z A Az z Where h is the depth to an element area da of the surface. The vertical component of force acting on a curved surface is equal to the net weight of the total column of fluid directl above the curved surface and has a line of action through the centroid of the fluid volume.

12 ME:56 Chapter Professor Fred Stern Fall 7 Eample Drum Gate hr-rcosθr(-cosθ) p γh γ R cosθ ( ) h n sinθ i+ cosθk da lrdθ π ( cosθ )( sinθiˆ + cosθkˆ ) lrd da F γr p π ( cosθ ) θdθ F.ˆ i F γlr sin γlr cosθ γrrl A p π + n θ π cos θ γlr 4 Same force as that on projection of gate onto vertical plane perpendicular direction

13 ME:56 Chapter Professor Fred Stern Fall 7 3 FF zz γγγγrr ππ ( cccccccc)cccccccccccc γγγγrr ssssssss θθ 4 sinθ ππ γγγγrr ππ γγγγ ππrr γγ Net weight of water above curved surface Another approach: FF γγγγ RR 4 ππrr γγγγrr 4 ππ FF γγγγ ππrr + FF FF FF FF γγγγγγrr

14 ME:56 Chapter Professor Fred Stern Fall 7 4 Hdrostatic Forces in Laered Fluids See tetbook.7 Buoanc and Stabilit Archimedes Principle F B F V ( ) F V () fluid weight above ABC fluid weight above ADC weight of fluid equivalent to the bod volume In general, FB ρg ( submerged volume). The line of action is through the centroid of the displaced volume, which is called the center of buoanc.

15 ME:56 Chapter Professor Fred Stern Fall 7 5 Eample: Floating bod in heave motion L G ddraft ρ b B h b W ρ Weight of the block b Lb hg mg γ where is A displaced water volume b the block and γ is the specific weight of the liquid. W ρb ρw ρb > ρw ρ < ρ b ρb B ρblbhg ρwlbdg d ρ w : d : d : d W B h > h sink < h floating Instantaneous displaced water volume: V A wp.. F m B W γ γ γ A wp wp w h S b h

16 ME:56 Chapter Professor Fred Stern Fall m+ γ A A wp wp.. γ + m Acosω t+ Bsinω t Use initial condition ( and B: Where n n.. t, ) to determine A cosωn sin ωn. t+ ω t ω n γ A m wp n period T π m π ω γa Spar Buo wp T is tuned to decrease response to ambient waves: we can increase T b increasing block mass m and/or decreasing waterline area A wp.

17 ME:56 Chapter Professor Fred Stern Fall 7 7 Stabilit of Immersed and Floating Bodies Here we ll consider transverse stabilit. In actual applications both transverse and longitudinal stabilit are important. Immersed Bodies Stable Neutral Unstable Static equilibrium requires: and M M requires that the centers of gravit and buoanc coincide, i.e., C G and bod is neutrall stable F v If C is above G, then the bod is stable (righting moment when heeled) If G is above C, then the bod is unstable (heeling moment when heeled)

18 ME:56 Chapter Professor Fred Stern Fall 7 8 Floating Bodies For a floating bod the situation is slightl more complicated since the center of buoanc will generall shift when the bod is rotated depending upon the shape of the bod and the position in which it is floating. Positive GM Negative GM The center of buoanc (centroid of the displaced volume) shifts laterall to the right for the case shown because part of the original buoant volume AOB is transferred to a new buoant volume EOD. The point of intersection of the lines of action of the buoant force before and after heel is called the metacenter M and the distance GM is called the metacentric height. If GM is positive, that is, if M is above G, then the ship is stable; however, if GM is negative, the ship is unstable.

19 ME:56 Chapter Professor Fred Stern Fall 7 9 α small heel angle CC lateral displacement of C C center of buoanc i.e., centroid of displaced volume V Solve for GM: find using () basic definition for centroid of V; and () trigonometr Fig. 3.7 () Basic definition of centroid of volume V V dv i Vi moment about centerplane V moment V before heel moment of V AOB + moment of V EOD due to smmetr of original V about ais i.e., ship centerplane V ( )dv + dv tan α / AOB EOD dv da tan α da V tan da α + tanαda AOB EOD

20 ME:56 Chapter Professor Fred Stern Fall 7 V tan α da ship waterplane area moment of inertia of ship waterplane about z ais O-O; i.e., I OO I OO moment of inertia of waterplane area about centerplane ais () Trigonometr V tan αi CC OO tan αi V OO CM tan α CM I OO / V GM CM CG GM I OO V CG GM > GM < Stable Unstable

21 ME:56 Chapter Professor Fred Stern Fall 7 Roll: The rotation of a ship about the longitudinal ais through the center of gravit. Consider smmetrical ship heeled to a ver small angle θ. Solve for the subsequent motion due onl to hdrostatic and gravitational forces. ( cos θˆj sinθiˆ ) ρ F b g M r M g g F b ( GCj ˆ + CC iˆ ) ( cosθˆj sinθiˆ ) ( GC sinθ + CC cosθ ) ( GC + GM CM ) sinθ kˆ GM sinθ kˆ kˆ ( ρ g Δ) Note: recall that M o F d, where d is the perpendicular distance from O to the line of action of F. M G GZ GM sinθ d O F

22 ME:56 Chapter Professor Fred Stern Fall 7 M G.. I θ I mass moment of inertia about long ais through G.. θ angular acceleration.. I θ+ GM sinθ.. GM for small θ : θ + θ I GM ρg GM mggm I I I k I definition of radius of gration m k I m GM I ggm k mk I The solution to this equation is,. θo θ ( t) θ cosω t + sinω t o n n ωn where θ the initial heel angle o for no initial velocit ω n natural frequenc ggm k ggm k

23 ME:56 Chapter Professor Fred Stern Fall 7 3 Simple (undamped) harmonic oscillation: The period of the motion is T π ω n T πk ggm Note that large GM decreases the period of roll, which would make for an uncomfortable boat ride (high frequenc oscillation). Earlier we found that GM should be positive if a ship is to have transverse stabilit and, generall speaking, the stabilit is increased for larger positive GM. However, the present eample shows that one encounters a design tradeoff since large GM decreases the period of roll, which makes for an uncomfortable ride.

24 ME:56 Chapter Professor Fred Stern Fall 7 4 Parametric Roll: The periodicit of the encounter wave causes variations of the metacentric height i.e. GMGM (t). Therefore:.. I θ+ GM () t θ Assuming GM ( t) GM + GMcos( ωt) :.. I θ+ ( GM + GMcos( ωt) ) θ.. θ+ ω + Cω cos( ω t) θ where ( n n e ) ω ggm GM ; ; ; ; and encounter wave freq. n C mg I mk ωe k GM B changing of variables ( τ ω ):.. θτ ( ) + δ ( + C cos τ ) θτ ( ) and e t ωn δ ωe This ordinar nd order differential equation where the restoring moment varies sinusoidall, is known as the Mathieu equation. This equation gives unbounded solution (i.e. it is unstable) when ωn n + δ n,,,3,.. ωe For the principle parametric roll resonance, n i.e. π π ω ω T T T T e n n e e n

25 ME:56 Chapter Professor Fred Stern Fall 7 5 Case () Rigid Bod Translation or Rotation In rigid bod motion, all particles are in combined translation and/or rotation and there is no relative motion between particles; consequentl, there are no strains or strain rates and the viscous term drops out of the N-S equation ( µ V ). p ρ ( g a) from which we see that p and lines of constant pressure must be perpendicular to this direction (b definition, f is perpendicular to f constant). acts in the direction of ( a) g, Rigid bod of fluid translating or rotating

26 ME:56 Chapter Professor Fred Stern Fall 7 6 The general case of rigid bod translation/rotation is as shown. If the center of rotation is at O where V V, the velocit of an arbitrar point P is: V V + Ω r where Ω the angular velocit vector and the acceleration is: dv dt a dv dω + Ω ( Ω r ) + r dt dt 3 acceleration of O centripetal acceleration of P relative to O 3 linear acceleration of P due to Ω Usuall, all these terms are not present. In fact, fluids can rarel move in rigid bod motion unless restrained b confining walls.

27 ME:56 Chapter Professor Fred Stern Fall 7 7.) Uniform Linear Acceleration pconstant ( g a) p ρ Constant p ρ a. a < p increase in +. a > p decrease in + p ρ + z ( g a ) z. a z > p decrease in +z. az < and az < g p decrease in +z but slower than g 3. a and a g z < > p increase in +z z ρ [( ) ] ^ ^ g + a k+ a i z

28 ME:56 Chapter Professor Fred Stern Fall 7 8 unit vector in the direction of p: s p p g+ a k+ ai ( ) z ( ) z g+ a + a lines of constant pressure are perpendicular to p. n s j a k g a i ( + ) z + ( + z) a g a unit vector in direction of pconstant angle between n and aes: θ tan ( g a + a z ) In general the pressure variation with depth is greater than in ordinar hdrostatics; that is: dp ds ( ) z p s ρ a + g+ a which is > ρg G p ρgs + constant ρgs gage pressure

29 ME:56 Chapter Professor Fred Stern Fall 7 9 ). Rigid Bod Rotation Consider a clindrical tank of liquid rotating at a constant rate Ω Ω k : p ρ ( g a) a Ω rω ( Ω r ) ^ er p ρ ( g a) gkˆ ρ + ρrω eˆ r i.e. and p ρr Ω r ρ p r Ω + f ( z) + c ρ p r Ω ρgz + Constant p ρ g z p z f ' ρg f ( z) ρgz + C The constant is determined b specifing the pressure at one point; sa, p p at (r,z) (,). ρ p p ρ gz + r Ω (Note: Pressure is linear in z and parabolic in r)

30 ME:56 Chapter Professor Fred Stern Fall 7 3 Curves of constant pressure are given b: p p ρ g r Ω g z + a + br which are paraboloids of revolution, concave upward, with their minimum points on the ais of rotation. The unit vector in the direction of p is: ρgkˆ + ρrω eˆ r sˆ ( ρg) + ( ρrω ) / [ ] z i.e. r C dz tanθ dr Ω g dz Ω z g g rω dr r Ω z g slope of s ln r ep equation of p surfaces The position of the free surface is found, as it is for linear acceleration, b conserving the volume of fluid. θ r s

31 ME:56 Chapter Professor Fred Stern Fall 7 3 Case (3) Pressure Distribution in Irrotational Flow; Bernoulli Equation Navier-Stokes: ρa ( p) ρgkˆ + µ V ρ t + V V V ( p + γz) + µ ( p + γz) + µ [ ( V ) ( V )] Viscous term for ρconstant and ω, i.e., Potential flow solutions also solutions NS under such conditions!. Assuming inviscid flow: µ Using vector identit VV VV V VV VV VV VV ρρ VV + VV VV VV VV (p + γz) Euler Equation V t. Assuming inviscid and incompressible flow: µ, ρconstant V + + p + gz V ω V V V ρ (ωω )

32 ME:56 Chapter Professor Fred Stern Fall Assuming inviscid, incompressible and stead flow: µ, ρconstant, t B V V B ω p + + ρ gz Consider: B perpendicular B constant V ω perpendicular V and ω Therefore, Bconstant contains streamlines and vorte lines:

33 ME:56 Chapter Professor Fred Stern Fall Assuming inviscid, incompressible, stead and irrotational flow: µ, ρconstant,, ω B B constant (everwhere same constant) V V t 5. Unstead inviscid, incompressible, and irrotational flow: µ, ρconstant, ω ϕ ϕ ϕ ϕ ϕ ϕ p gz t ρ ϕ ϕ ϕ p gz B( t) t ρ B(t) time dependent constant

34 ME:56 Chapter Professor Fred Stern Fall 7 34

35 ME:56 Chapter Professor Fred Stern Fall 7 35 Larger speed/densit or smaller R require larger pressure gradient or elevation gradient normal to streamline.

36 ME:56 Chapter Professor Fred Stern Fall 7 36 Flow Patterns: Streamlines, Streaklines, Pathlines ) A streamline is a line everwhere tangent to the velocit vector at a given instant. ) A pathline is the actual path traveled b a given fluid particle. 3) A streakline is the locus of particles which have earlier passed through a particular point. Note:

37 ME:56 Chapter Professor Fred Stern Fall For stead flow, all 3 coincide.. For unstead flow, ψ(t) pattern changes with time, whereas pathlines and streaklines are generated as the passage of time. Streamline B definition we must have V dr which upon epansion ields the equation of the streamlines for a given time t t d u d dz ds s integration parameter v w So if (u,v,w) known, integrate with respect to s for tt with IC (,,z,t) at s and then eliminate s. Pathline The pathline is defined b integration of the relationship between velocit and displacement. d dt u d dt v dz dt w Integrate u,v,w with respect to t using IC ( eliminate t.,, z, t ) then Streakline

38 ME:56 Chapter Professor Fred Stern Fall 7 38 To find the streakline, use the integrated result for the pathline retaining time as a parameter. Now, find the integration constant which causes the pathline to pass through (,, z ) for a sequence of times ξ < t. Then eliminate ξ. Eample: b: an idealized velocit distribution is given u v w + t + t calculate and plot: ) the streamlines ) the pathlines 3) the streaklines which pass through (,, z ) at t..) First, note that since w there is no motion in the z direction and the flow is -D d ds + t C s at ep( and eliminating s d ds + t s s ) C ep( ) + t + t (, ) : C C

39 ME:56 Chapter Professor Fred Stern Fall 7 39 s ( + t)ln ( + t)ln n + t ( ) where n + t This is the equation of the streamlines which pass through (, ) for all times t..) To find the pathlines we integrate d d dt + t dt + t C ( + t) C ( + t) t (, ) (, ) : C C now eliminate t between the equations for (, ) [ + ( )] This is the pathline through (, ) at t and does not coincide with the streamline at t.

40 ME:56 Chapter Professor Fred Stern Fall ) To find the streakline, we use the pathline equations to find the famil of particles that have passed through the point (, ) for all times t < ξ. : ] ) )( [( ) ( ) )( ( ) ( ) ( ) ( ) ( t t t t t C C t C t C ξ ξ ξ

41 ME:56 Chapter Professor Fred Stern Fall 7 4 The Stream Function Powerful tool for -D flow in which V is obtained b differentiation of a scalar ψ which automaticall satisfies the continuit equation. Note for D flow VV,, (,, ωω zz) boundar conditions (4 required): at infinit : on bod : u ψ U u v ψ v ψ ψ

42 ME:56 Chapter Professor Fred Stern Fall 7 4 Irrotational Flow ψ on S : on S : B nd order linear Laplace equation ψ U + const. ψ const. u ψ φ v ψ φ Ψ and φ are orthogonal. dφ φ d + φ d ud + vd dψ ψ d + ψ d vd + ud d u i.e. d φ const v d dψ const

43 ME:56 Chapter Professor Fred Stern Fall 7 43 Geometric Interpretation of ψ Besides its importance mathematicall ψ also has important geometric significance. ψ constant streamline Recall definition of a streamline: V dr compare with i.e. d d u v ud vd dψ dr diˆ + dj ˆ dψ ψ d + ψ d along a streamline vd + ud Or ψ constant along a streamline and curves of constant ψ are the flow streamlines. If we know ψ (, ) then we can plot ψ constant curves to show streamlines.

44 ME:56 Chapter Professor Fred Stern Fall 7 44 Phsical Interpretation dq V. nda ˆ ψ ( ˆ ψ d ).( ˆ d i j i ˆj ) ds ds ds ψ d + ψ d dψ (note that ψ and Q have same dimensions: m 3 /s) (Not Streamline) i.e. change in ψ d is volume flu and across streamline Q. ψ ψ ψ V nda d Consider flow between two streamlines: dq.

45 ME:56 Chapter Professor Fred Stern Fall 7 45 Incompressible Plane Flow in Polar Coordinates continuit : ( rvr ) + ( vθ ) r r r θ or : ( rvr ) + ( vθ ) r θ ψ ψ sa: vr vθ r θ r ψ ψ then ( r ) + ( ) r r θ θ r as before dψ along a streamline and dq dψ volume flu change in stream function Incompressible aismmetric flow ( ) ( ) rv + v continuit : r r r z z ψ ψ sa : v v r r z z r r ψ ψ then : r + r r r z z r r as before dψ along a streamline and dq dψ

46 ME:56 Chapter Professor Fred Stern Fall 7 46 Generalization Stead plane compressible flow: constant isa streamline and i.e. ) ( compare with Alongside compressibleflow stream function define : ) ( ) ( ψ ψ ψ ρ ψ ψ ψ ψ ρ ψ ρ ψ ψ ψ ρ ψ ρ ρ ρ d d d d d d d vd ud v u v u Now: ). ( ). ( ψ ψ ρ ψ ρ da n V m d da n V dm Change in ψ is equivalent to the mass flu.