57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 9.5 Qantitative Relatins r the Trblent Bndary ayer Descriptin Trblent Flw V and p are randm nctins time in a trblent lw The mathematical cmpleity trblence entirely precldes any eact analysis. A statistical thery is well develped; hwever, it is bth beynd the scpe this crse and nt generally sel as a predictive tl. Since the time Reynlds (88) trblent lws have been analyzed by cnsidering the mean (time averaged) mtin and the inlence trblence n it; that is, we separate the velcity and pressre ields int mean and lctating cmpnents p p p v v v and r cmpressible lw w w w ρ ρ ρ and T T T
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 4 where (r eample) t t dt t t and t siciently large that the average is independent time Ths by deinitin, etc. Als, nte the llwing rles which apply t tw dependent variables and g g g g g s s ds ds (, v, w, p) s (, y, z, t) The mst imprtant inlence trblence n the mean mtin is an increase in the lid stress de t what are called the apparent stresses. Als knwn as Reynlds stresses: τ ij ρ i j ρ ρ v ρ w ρ v ρv ρv w ρ w ρv w ρw Symmetric nd rder tensr
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 5 The mean-lw eqatins r trblent lw are derived by sbstitting V V V int the Navier-Stkes eqatins and averaging. The reslting eqatins, which are called the Reynlds-averaged Navier-Stkes (RANS) eqatins are: Cntinity V i.e. V and V DV ρ i j gkˆ Dt j Mmentm ( ) ρ p µ V r ρ ρ DV Dt ρgkˆ p τ v j i τ ij µ ρ i j w j i Cmments: τ ij ) eqatins are r the mean lw ) dier rm laminar eqatins by Reynlds stress terms i j ) inlence trblence is t transprt mmentm rm ne pint t anther in a similar manner as viscsity 4) since i j are nknwn, the prblem is indeterminate: the central prblem trblent lw analysis is clsre! 4 eqatins and 4 6 nknwns ij y z
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 6
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 7 -D Bndary-layer Frm RANS eqatins v y p v y ρ ( v ) e ν y y reqires mdeling Trblence Mdeling Clsre the trblent RANS eqatins reqire the determinatin ρ v, etc. Histrically, tw appraches were develped: (a) eddy viscsity theries in which the Reynlds stresses are mdeled directly as a nctin lcal gemetry and lw cnditins; and (b) mean-lw velcity prile crrelatins which mdel the mean-lw prile itsel. The mdern appraches, which are beynd the scpe this class, invlve the sltin r transprt PDE s r the Reynlds stresses which are slved in cnjnctin with the mmentm eqatins. (a) eddy-viscsity: theries (mainly sed with dierential methds) v In analgy with the laminar viscs ρ µ t y stress, i.e., τ t mean-lw rate strain The prblem is redced t mdeling µ t, i.e., µ t µ t (, lw at hand) Varis levels sphisticatin presently eist in mdeling µ t
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 8 µ t ρv trblent velcity scale t t trblent length scale where V t and t are based an large scale trblent mtin The ttal stress is τ ttal mleclar viscsity ( µ µ ) y t eddy viscsity (r high Re lw µ t >> µ) Miing-length thery (Prandtl, 9) ρ v cρ v v y y based n kinetic thery gases and are miing lengths which are analgs t mleclar mean ree path, bt mch larger ρ v ρ y y Knwn as a zer eqatin mdel since n additinal PDE s are slved, nly an algebraic relatin ( y) distance acrss shear layer (bndary layer, jet, wake, etc.)
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 9 Althgh miing-length thery has prvided a very sel tl r engineering analysis, it lacks generality. Therere, mre general methds have been develped. One and tw eqatin mdels µ C cnstant t Cρk ε k trblent kinetic energy v w ε trblent dissipatin rate Gverning PDE s are derived r k and ε which cntain terms that reqire additinal mdeling. Althgh mre general then the zer-eqatin mdels, the k-ε mdel als has deinite limitatin; therere, recent wrk invlves the sltin PDE s r the Reynlds stresses themselves. Diiclty is that these cntain triple crrelatins that are very diiclt t mdel.
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 (b) mean-lw velcity prile crrelatins (mainly sed with integral methds) As an alternative t mdeling the Reynlds stresses ne can mdel mean lw prile directly. Fr simple -D lws this apprach is qite d and will be sed in this crse. Fr cmple and -D lws generally nt sccessl. Cnsider the shape trblent velcity priles. Nte that very near the wall τ laminar mst dminate since ρ i j at the wall (y ) and in the ter part trblent stress will dminate. This leads t the three layer cncept: Inner layer: Oter layer: viscs stress dminates trblent stress dminates Overlap layer: bth types stress imprtant
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 ) Inner layer (Prandtl, 9) Frm dimensinal analysis (µ, τ w, ρ, y) nte: nt () ( ) y law--the-wall y where: rictin velcity τ w / ρ y y ν very near the wall: τ τ w cnstant ) Oter layer (Karmen, 9) d µ cy r y dy ( ) g(, τ, ρ, y) e ter dp nte: independent µ and actally als depends n w Frm dimensinal analysis e y velcity deect law
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 ) Overlap layer (Milliken, 97) In rder r the inner and ter layers t merge smth ln y κ B lg-law.4 5 κ and B rm eperiments and independent dp/
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 Nte that the y scale is lgarithmic and ths the inner law nly etends ver a very small prtin Inner law regin <. And the lg law encmpasses mst the bndary-layer. Ths as an apprimatin ne can simply assme ln y κ B y ν is valid all acrss the shear layer. This is the apprach sed in this crse r trblent lw analysis. The apprach is a gd apprimatin r simple and -D lws (pipe and lat plate), bt des nt wrk r cmple and -D lws. y τ w / ρ
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 4
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 5 Mmentm Integral Analysis Backgrnd: Histry and Mdern Apprach: FD T btain general mmentm integral relatin which is valid r bth laminar and trblent lw Fr lat plate r r general case ( mmentm eqatin ( v)cntinity)dy y τ w ρ c dθ ( H) d θ d dp ρ d lat plate eqatin θ dy mmentm thickness H shape parameter θ dy displacement thickness Can als be derived by CV analysis as shwn net r lat plate bndary layer.
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 6 Mmentm Eqatin Applied t the Bndary ayer y h streamline starts in nirm lw merges with at Steady ρ cnstant neglect g v << p cnstant i.e., - p CV,,, 4 D drag b τw pressre rce r v << rce n CV wall shear stress F D ρ ρ ( V da) ρ ( V da) bh ρb ( ) dy D () ρ bh ρb dy
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 7 net eliminate h sing cntinity ρ V da ρ V da ρ h dy bh ρb dy depends n (y) D ( ) ρb dy ρb b ρ ( ) dy dy C D D ρ b dy θ mmentm thickness C D θ
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 8 b b A D C w D θ ρ τ ρ ( ) ( ) θ ρ τ w d w θ ρ τ d c θ c lcal skin rictin ceicient mmentm integral relatin r lat plate bndary layer θ dy
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 9 Apprimate sltin r a laminar bndary-layer Assme cbic plynmial r (y) Dy Cy By A y y A B y ; y C D i.e., y y d c w θ ρ τ mmentm integral eqatin r dp d.9 µ ρ dy θ y y y dy d w µ τ
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 4 i.e., 4.65 Re Cmpare with Eact Blassis 5 7% Re τ w. ρv Re. ρ Re % c C.646 Re.9 Re.664 Re. Re C ρ τ b w ( ) span length ttal skin-rictin drag ceicient
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 4 Apprimate sltin Trblent Bndary-ayer Re t X 6 c dθ r a lat plate bndary layer Re crit 5, as was dne r the apprimate laminar lat plate bndary-layer analysis, slve by epressing c c () and θ θ() and integrate, i.e. assme lg-law valid acrss entire trblent bndary-layer y ln κ ν B neglect laminar sb layer and velcity deect regin at y, ln κ ν B Re c / / / c r.44ln Re 5 c c () c. Re / 6 pwer-law it
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 4 Net, evalate dθ d dy can se lg-law r mre simply a pwer law it / 7 y Nte: can nt be sed t btain c () since τ w 7 θ 7 θ( ) τ w c ρ ρ dθ 7 7 ρ d r Re.6 Re d 9.7 / 6 / 7 6/ 7 almst linear i.e., mch aster grwth rate than laminar bndary layer c.7 Re / 7 C. / 7 Re 7 6 C ( )
57: Mechanics Flids and Transprt Prcesses Chapter 9 Pressr Fred Stern Typed by Stephanie Schrader Fall 999 4 Alternate rms given in tet depending n eperimental inrmatin and pwer-law it sed, etc. (i.e., dependent n Re range.) Sme additinal relatins given in tets r larger Re are as llws: Ttal.455 7 shear-stress C Re > 7.58 ( lg Re ) Re ceicient c (.98lg Re.7) cal shear-stress ceicient c. ( lg Re. ) 65 Finally, a cmpsite rmla that takes int accnt bth the initial laminar bndary-layer (with translatin at Re CR 5,) and sbseqent trblent bndary layer.74 7 is C 5 < Re < 7 /5 Re Re