Turbulent Flow. Turbulent Flow

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6 1. Caothc fluctuatons wth a wde range of frequences and ampltudes. Unsteady flow of aleatory nature. For engnnerng purposes, t s often requred to apply statstcal methods (average values, standard devatons, spatal and/or tme correlatons) 2. Three-dmensonal flows. Hghly dstorted eddes of dfferent shapes and dmensons

7 3. Hgh mxng propertes. Large heat, momentum and mass transfer rates, whch can be orders of magntude larger than those due to molecular dffuson. 4. Turbulent flow s hghly dsspatve. Energy s transferred from the mean flow to the turbulence feld (fluctuatons) from the largest eddes. Energy transfer process (small to hgh frequences) due to vortex stretchng. 1. Aleatory 1. Three-dmensonal 1.Hgh dffuson 1. Dsspatve 1.Property of the flow 1.Contnuous medum 1.Hgh Reynolds number

8 Energy cascade wth frequency or wavenumber Smallest frequences, largest wavelengths, correspond to the eddes wth the hghest turbulence knetc energy. The largest dmensons of the eddes s lmted by the boundary condtons. Hghly anysotropc eddes. Energy cascade wth frequency or wavenumber The hghest frequences, lowest wavelengths, correspond to the dsspatve eddes. Smallest dmensons of the eddes s lmted by the molecular shear-stress. Smallest Reynolds numbers, hgher vscous effects, mply an ncrease of the dmensons of the dsspatve eddes. Isotropc eddes.

9 Energy cascade wth frequency or wavenumber Inertal range n the ntermedate regon promotng the energy transfer by a mechansm nvolvng vortex stretchng Entranment A turbulent boundary-layer grows by molecular dffuson and entranment,.e. entranng external flud n to the boundary-layer. The entranment effect s sgnfcantly larger than molecular dffuson.

10 Entranment For zero pressure gradent Lamnar boundary-layer grows approxmately 2.5mm per meter Turbulent boundary-layer grows about 18mm each meter Entranment Entranment velocty, V E quantfes the ncrease of the volumetrc flow rate along the boundary-layer dq d δ d VE = = udy = dx dx 0 dx 0 δ U e U e * [ U ( δ )] u d 1 dy = e δ U e dx

11 Coanda effect Transton Inflecton pont Separaton Recrculaton bubble Re-attachment Re-attachment of a shear-layer to a nearby sold wall Coanda effect Transton Inflecton pont Separaton Recrculaton bubble Re-attachment Lamnar free shear-layer ncludes an nflecton pont. Therefore, there s a quck transton to turbulent flow

12 Coanda effect Transton Inflecton pont Separaton Recrculaton bubble Re-attachment Entranment of the turbulent shear-layer dffuses momentum to the flud close to the step makng the pressure drop. Coanda effect Transton Inflecton pont Separaton Recrculaton bubble Re-attachment Transverse pressure gradent deflects the free shear-layer to the wall to balance the forces

13 Drect Numercal Smulaton, DNS Naver-Stokes equatons solved numercally wth a grd spacng and a tme step suffcently small to resolve the smallest eddes of the flow. On the other hand, smulaton tme must be large enough to capture the effects of the largest eddes Numercal accuracy of the soluton s very mportant (to avod msleadng results due to numercal dffuson) Dependent varables change n space and tme,.e. they are the nstataneous values at a gven flow locaton Large-Eddy Smulaton, LES Naver-Stokes equatons fltered n space. Extra mathematcal model requred to nclude the effect of the fltered scales. Tme dependent numercal soluton. Numercal accuracy s also mportant ( on-gong debate about how much ). Correct applcaton to near-wall flows s dffcult. Dependent varables change wth tme, but have a dfferent meanng of DNS, due to flterng n space

14 Mult-Scale Smulaton Naver-Stokes equatons splt n to small and large scales. Analytcal methods used to obtan an approxmate soluton of the small scales. Numercal soluton of the large scales n tme wth effect of small scales ncluded from ts approxmate soluton. Numercal accuracy s stll mportant. Applcaton nearwalls troublesome ( small scales become too small...) Reynolds-averaged equatons Statstcs appled to mass conservaton and momentum balance (contnuty and Naver-Stokes). Type of statstc handlng depends on flow propertes: 1. Spatal averagng 2. Tme averagng 3. Ensemble averagng Instataneous velocty components, u ~, (dependent varables) splt n to a mean value, U, and a fluctuaton, u u ~ = U + u

15 Reynolds-averaged equatons 1. Spatal averagng U j n = 1 = lm n u ~ j ( x, y, z ) n Homogeneous turbulence Reynolds-averaged equatons 2. Tme averagng U = lm T t o T + ~ to T u dt Statstcally steady flow

16 Reynolds-averaged equatons 3. Ensemble averagng U j = 1 = lm n n ( u ~ j ( t ) ) n Mean values are tme dependent. Sutable for perodc flows.