Primary dependent variable is fluid velocity vector V = V ( r ); where r is the position vector

Transcription

1 Chapter 4: Flids Kinematics 4. Velocit and Description Methods Primar dependent ariable is flid elocit ector V V ( r ); where r is the position ector If V is known then pressre and forces can be determined sing techniqes to be discssed in sbseqent chapters. r r î ĵ kˆ V (r,t) î ĵ wkˆ Consideration of the elocit field alone is referred to as flow field kinematics in distinction from flow field dnamics (force considerations). Flid mechanics and especiall flow kinematics is a geometric sbject and if one has a good nderstanding of the flow geometr then one knows a great deal abot the soltion to a flid mechanics problem. Consider a simple flow sitation, sch as an airfoil in a wind tnnel: U constant

2 Velocit: Lagrangian and Elerian Viewpoints There are two approaches to analing the elocit field: Lagrangian and Elerian Lagrangian: keep track of indiidal flids particles (i.e., sole F Ma for each particle) Sa particle p is at position r (t ) and at position r (t ) then, r r d d d V î ĵ kˆ p lim t 0 t t î ĵ w kˆ p p p Of corse the motion of one particle is insfficient to describe the flow field, so the motion of all particles mst be considered simltaneosl which wold be a er difficlt task. Also, spatial gradients are not gien directl. Ths, the Lagrangian approach is onl sed in special circmstances. Elerian: focs attention on a fied point in space î ĵ kˆ In general, V V(, t) î ĵ wkˆ elocit components

3 3 where, (,,,t), (,,,t), w w(,,,t) This approach is b far the most sefl since we are sall interested in the flow field in some region and not the histor of indiidal particles. Howeer, mst transform F Ma from sstem to CV (recall Renolds Transport Theorem (RTT) & CV analsis from thermodnamics) E. Flow arond a car V can be epressed in an coordinate sstem; e.g., polar or spherical coordinates. Recall that sch coordinates are called orthogonal crilinear coordinates. The coordinate sstem is selected sch that it is conenient for describing the problem at hand (bondar geometr or streamlines). V ê ê r r θ θ r cos θ r sin θ ê r êθ cos θ î sin θ ĵ sin θ î cos θ ĵ Undobtedl, the most conenient coordinate sstem is streamline coordinates: V(s,t) (s, t)ê (s,t) s s Howeer, sall V not known a priori and een if known streamlines mabe difficlt to generate/determine.

4 4 4. Flow Visaliation and Plots of Flid Flow Data See tetbook for: Streamlines and Streamtbes Pathlines Streaklines Timelines Refractie flow isaliation techniqes Srface flow isaliation techniqes Profile plots Vector plots Contor plots 4.3 Acceleration Field and Material Deriatie The acceleration of a flid particle is the rate of change of its elocit. In the Lagrangian approach the elocit of a flid particle is a fnction of time onl since we hae described its motion in terms of its position ector.

5 5 r V a a p p p p (t)î p (t)ĵ p (t)kˆ drp pî p ĵ w pkˆ dp d rp a î a ĵ a kˆ d p dp dw p a a In the Elerian approach the elocit is a fnction of both space and time; conseqentl, V (,,,t)î (,,,t)ĵ w(,,,t)kˆ a dv d î d ĵ dw kˆ a î a ĵ a kˆ,, are f(t) since we mst follow the particle in ealating d/ a d t t t t t w D called sbstantial deriatie Dt Similarl for a & a, a a D Dt Dw Dt w t w t w w w w

6 6 In ector notation this can be written concisel DV V V V Dt t î ĵ kˆ gradient operator V First term,, called local or temporal acceleration reslts t from elocit changes with respect to time at a gien point. Local acceleration reslts when the flow is nstead. Second term, V V, called conectie acceleration becase it is associated with spatial gradients of elocit in the flow field. Conectie acceleration reslts when the flow is non-niform, that is, if the elocit changes along a streamline. The conectie acceleration terms are nonlinear which cases mathematical difficlties in flow analsis; also, een in stead flow the conectie acceleration can be large if spatial gradients of elocit are large. Eample: Flow throgh a conerging nole can be approimated b a one dimensional elocit distribtion (). For the nole shown, assme that the elocit aries linearl from V o at the entrance to 3V o at the eit. Compte the acceleration DV as a fnction of. Dt

7 7 DV Ealate Dt at the entrance and eit if Vo 0 ft/s and L ft. () m b We hae D V () î, a Dt (0) b V o 3V m o V L o V L o Assme linear ariation between inlet and eit () Vo o o L L ( ) V V V0 Vo a L L 0 a 00 L a 600 ft/s

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12 Eample problem: Deformation rate of flid element Consider the stead, two-dimensional elocit field gien b V (, ) ( ) i (.5 0.8) j Calclate the kinematic properties sch as; (a) Rate of translation (b) Rate of rotation (c) Rate of linear strain (d) Rate of shear strain Soltion: (a) Rate of translation: ,.5 0.8, w 0 (b) Rate of rotation: ω k ( 0 0 ) k 0 (c) Rate of linear strain: ε 0.8s, ε 0.8s, ε 0 (d) Rate of shear strain: ε ( 0 0 ) 0

13 3 Rotation and Vorticit Ω flid orticit anglar elocit ω V i.e., crl V ω w kˆ ĵ î kˆ ĵ w î w To show that this definition is correct consider two lines in the flid

14 4 Anglar elocit abot ais aerage rate of rotation d d d tan d d d α β α ω lim 0 i.e., d α d d d tan d β lim 0 i.e., d β similarl, ω ω ω w w i.e., Ω ω

15 5 Eample problem: Calclation of Vorticit Consider the following stead, three-dimensional elocit field V (,, w) (3.0.0 ) i (.0.0 ) j ( 0.5) k Calclate the orticit ector as a fnction of space (,, ) Soltion: Vorticit ector in Cartesian coordinates: w w ζ i j k For 3.0.0,.0.0, w 0.5 ζ i j k ( ) ( 0.5 0) ( ).0 ( ) ( 0.5i ) ( 0.5) j ( 3.0) k