2. Fluid-Flow Equations

Size: px

Start display at page:

Download "2. Fluid-Flow Equations"

Madlyn Mills
6 years ago
Views:

1 . Fli-Flo Eqation Governing Eqation Conervation eqation for: ma momentm energy (other contitent) Alternative form: integral (control-volme) eqation ifferential eqation Integral (Control-olme) Approach Conier the get of any phyical qantity in any control volme RATE OF CHANGE inie ADECTION throgh onaryof DIFFSION SORCE inie Finite-volme metho for CFD 1

2 Ma Conervation (Continity) Ma conervation: ma i neither create nor etroye (ma) net inar ma flx (ma) net otar ma flx A n (ma) face (ma flx) Ma in a cell: ρ Ma flx throgh a face: C ρ A n ρ A Ma Conervation - Differential Eqation t (ma) net otar ma flx z n y e x (ρ ) (ρa) e (ρa) (ρva) n (ρva) (ρa) t (ρa) (ρδxδyδz ) [(ρ) e (ρ) ]ΔyΔz [(ρv) n (ρv) ]ΔzΔx [(ρ) t (ρ) ]ΔxΔy ρ (ρ) e (ρ) Δx (ρv) n (ρv) Δy (ρ) t (ρ) Δz ρ (ρ) (ρv) (ρ) t x y z ρ (ρ) t Continity in Incompreile Flo t (volme) net otarvolme flx z n y e x v x y z

3 Momentm Eqation Momentm Principle: force = rate of change of momentm F If teay: force = (momentm flx) ot (momentm flx) in If nteay: force = / (momentm inie control volme) + (momentm flx) ot (momentm flx) in Momentm Eqation Rate of change of momentm = force (momentm) net otarmomentm flx force A n (ma ) face (ma flx ) F Momentm of fli in a cell Momentm flx throgh a face = ma = ma flx ( ρ ) ( ρ A) Fli Force Srface force (proportional to area): prere y force tre area vico force: τ μ y Boy force (proportional to volme): z force enity force volme gravity: ρge z g axi R R centrifgal force: ρω R r Corioli force: ρω In inertial frame In rotating frame 3

4 Differential Eqation t (momentm) net momentm flx force z n y e (ρ) (ρa) e e (ρa) (ρva) n n (ρva) (ρa) t t (ρa) p A pe Ae vico an other force x (ρδxδyδz ) [(ρ) ee (ρ) ]ΔyΔz [(ρv) nn (ρv) ]ΔzΔx ( p pe)δyδz vico an other force [(ρ) t t (ρ) ]ΔxΔy (ρ) (ρ) e (ρ) (ρv) n (ρv) (ρ) t (ρ) Δx Δy Δz ( pe p) vico an other force Δx (ρ) (ρ) (ρv) (ρ) p μ other force t x y z x General Scalar Rate of change + net otar flx = orce Amont in a cell: Flx throgh a face: = concentration (amont per nit ma) (ma concentration) A n avection: iffion: ( ρ A) Γ A n (ma flx concentration) (iffivity graient area) Sorce: S (orce enity volme) (ma ) (ma flx Γ A) n face (ρ) (ρ Γ ) (ρv Γ ) (ρ Γ ) t x x y y z z Momentm Component a General Scalar Momentm eqation: (ma ) face ma flx (ma ) (ma flx μ A) other force n face (μ A) face n other force vico force General calar-tranport eqation: (ma ) (ma flx Γ A) n face S elocity component, v, atify inivial calar-tranport eqation: concentration, velocity component iffivity, vicoity orce, S other force Difference: momentm eqation are non-linear momentm eqation are cople the velocity fiel alo ha to e ma-conitent 4

5 Differential Eqation For Fli Flo Form of the eqation in primitive variale may e: Conervative can e integrate irectly to give net flx = orce Non-conervative can t e integrate irectly Other form of the eqation incle thoe for: Derive variale e.g. velocity potential; tream fnction. Example ( y ) g( x) x y y g( x) x conervative non-conervative Same eqation!... t only the firt can e integrate irectly Rate of Change Folloing the Flo t,x) Total erivative (folloing any path x(t) x y z t x y z (x(t), y(t), z(t)) Material erivative (folloing the flo): x, etc D v t x y z D t 5

6 Non-Conervative Flo Eqation conervative form non-conervative form D (ρ) (ρ) (ρv) (ρ) ρ t x y z ( ma conervation) D p ρ μ e.g. momentm eqation: x ma acceleration force Proof: (ρ) (ρ) (ρv) (ρ) t x y z ρ (ρ) (ρv) (ρ) ρ ρ ρv ρ t t x x y y z z ρ (ρ) (ρv) (ρ) t x y z D ρ ycontinity ρ v t x y z D/ yefinition Example Q1 (Eqation Maniplation) In - flo, the continity an x-momentm eqation can e ritten in conervative form a ρ p (ρ) (ρv) ( ρ) (ρ) (ρv) μ t x y t x y x (a) Sho that thee can e ritten in the eqivalent non-conervative form: Dρ v ρ( ) x y D p ρ μ x () Define careflly hat i meant y the tatement that a flo i incompreile. To hat oe the continity eqation rece in incompreile flo? (c) Write on conervative form of the 3- eqation for ma an x-momentm. () Write on the z-momentm eqation, incling the gravitational force. (e) Sho that, for contant-enity flo, prere an gravity can e comine in the momentm eqation via the piezometric prere p + gz. axi R R (f) In a rotating reference frame there are aitional apparent force (per nit volme): centrifgal force: ρω ( Ω r) or r ρω R Corioli force: ρω here Ω i the anglar velocity of the reference frame, i the fli velocity in that frame, r i the poition vector an R i it projection perpeniclar to the axi of rotation. By riting the centrifgal force a the graient of ome qantity ho that it can e me into a moifie prere. Alo, fin the component of the Corioli force if rotation i aot the z axi. Example Q (Exact Soltion) The x-component of the momentm eqation i given y D p ρ μ x ing thi eqation, erive the velocity profile in flly-evelope, laminar flo for: (a) () prere-riven flo eteen tationary parallel plane ( Plane Poieille flo ); contant-prere flo eteen tationary an moving plane ( Coette flo ). Ame flo in the x irection, an oning plane y = an y = h. The velocity i then ((y),,). y (y) h In part (a) oth all are tationary. In part () the pper all lie parallel to the loer all ith velocity. x 6

7 Non-Dimenionaliation - Avantage All ynamically-imilar prolem (ame Re etc.) can e olve ith a ingle comptation The nmer of parameter i rece It inicate the relative ize of ifferent term in the governing eqation: in particlar, hich might e neglecte Comptational variale are imilar ize, yieling etter nmerical accracy Non-Dimenionaliation Form non-imenional variale ing length (L ), velocity ( ) an enity ( ) cale: L,,, ρ ρ ρ x L, ρ x t t p pref p, etc. Stitte into the governing eqation: D p ρ μ x ρ D ρ p μ ρ L L x L D p μ ρ x ρ L D p ρ x 1 Re Ientify important imenionle grop: ρ Re μ L D p 1 ρ x Re Common Dimenionle Grop ρl Re μ Reynol nmer (vico flo; μ = ynamic vicoity) Fr gl Froe nmer (open-channel flo; g = gravity) Ma c Mach nmer (compreile flo; c = pee of on) Ro ΩL Roy nmer (rotating flo; Ω = anglar velocity of frame) ρ L We σ Weer nmer (free-rface flo; σ = rface tenion) 7

8 Smmary (1) The fli-flo eqation are conervation eqation for: ma momentm energy (aitional contitent) The eqation can e ritten in eqivalent integral (control-volme) or ifferential form The finite-volme metho i a irect icretiation of the control-volme eqation Differential form of the flo eqation may e conervative or nonconervative For any conerve property an aritrary control volme: rate of change + net otar flx = orce Smmary () There are really jt to canonical eqation to olve: ma conervation (continity) a generic calar-tranport eqation Each Carteian velocity component atifie it on calar-tranport eqation Hoever, the momentm eqation are: non-linear cople alo reqire to e ma-conitent Non-imenionaliation: olve ynamically-imilar (Re, Fr, Ro, ) flo ith a ingle comptation rece the nmer of parameter ientifie relative importance of ifferent term in eqation maintain nmerical variale of imilar ize 8

2. FLUID-FLOW EQUATIONS SPRING 2019

2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid