Fundamental Equations of Fluid Mechanics
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- Σοφοκλής Αγγελόπουλος
- 8 χρόνια πριν
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1 Fundamental Equations of Fluid Mechanics 1 Calculus 1.1 Gadient of a scala s The gadient of a scala is a vecto quantit. The foms of the diffeential gadient opeato depend on the paticula geomet of inteest. Below these foms ae povided in ectangula clindical pola and spheical pola coodinates. Rectangula coodinates s = s i s i s i z. 1 Clindical pola coodinates s = s i 1 s i s i z. 2 Spheical pola coodinates 1.2 Gadient of a vecto u s = s i 1 s i 1 s sin i. 3 The gadient of a vecto u is a tenso quantit. The diffeential foms of the gadient of a vecto ae povided below in ectangula clindical pola and spheical pola coodinates. Rectangula coodinates The vecto is given b u = u i u i u z i z so u is given b: Clindical pola coodinates u = u i i u i i u i zi u i i u i i u i zi i i z i i z i zi z. 4 The vecto is given b u = u i u i u z i z so u is given b: u = u i i u i i i i z 1 1 u 1 u 1 u u i i u i zi i i u i zi i i z i zi z. 5 1
2 Spheical pola coodinates The vecto is given b u = u i u i u i so u is given b: u = u i i u i i 1 1 u u u 1 i i u i i u i i 1 u i i 1.3 Divegence of a vecto u sin 1 u sin 1 u sin u u u cot cot u i i u i i i i. 6 The divegence of a vecto u is a scala quantit. The diffeential foms of the divegence depend on the paticula geomet of inteest. Below these foms ae povided in ectangula clindical pola and spheical pola coodinates. Rectangula coodinates The vecto is given b u = u i u i u z i z so u is given b: Clindical pola coodinates u = u u. 7 The vecto is given b u = u i u i u z i z so u is given b: Spheical pola coodinates u = 1 u 1 u. 8 The vecto is given b u = u i u i u i so u is given b: u = u 1.4 Divegence of a tenso T 1 sin u 1 u sin sin. 9 The divegence of a tenso T is a vecto quantit. The diffeential foms of the divegence depend on the paticula geomet of inteest. Below these foms ae povided in ectangula clindical pola and spheical pola coodinates. Rectangula coodinates The tenso is given b T = T i i T i i T z i i z T i i T i i T z i i z T z i z i T z i z i T zz i z i z so T is given b: 2
3 T = T T Tz T T T z T z T z T zz i i i z. 10 Clindical pola coodinates The tenso is given b T = T i i T i i T z i i z T i i T i i T z i i z T z i z i T z i z i T zz i z i z so T is given b: T = 1 T 1 T T z 1 2 T 2 1 T 1 T z 1 T z T zz T i T z 1 T T i i z. 11 Spheical pola coodinates The tenso is given b T = T i i T i i T i i T i i T i i T i i T i i T i i T i i so T is given b: T = 1 2 T 2 1 sin T 1 T sin sin 1 T T i 1 3 T 3 1 sin T 1 T sin sin 1 cot T T T i 1 3 T 3 1 sin T 1 T sin sin 1 cot T T T i Cul of a vecto u The cul of a vecto is also a vecto quantit. Once again the diffeential foms of the culs depend on the paticula geomet of inteest. Below these foms ae povided in ectangula clindical pola and spheical pola coodinates. Rectangula coodinates The vecto is given b u = u i u i u z i z so u is given b: u = uz u u i u z u i u i z. 13 3
4 Clindical pola coodinates The vecto is given b u = u i u i u z i z so u is given b: u = 1 u u i u z i Spheical pola coodinates 1 u 1 The vecto is given b u = u i u i u i so u is given b: u = 1 sin u 1 sin sin 1.6 Laplacian of a scala 2 s u 1 u i sin 1 u i z. 14 u i The Laplacian is fomed fom the divegence of a gadient opeato: = 2. The Laplacian of of a scala is a scala quantit. The foms of the diffeential Laplacian opeato depend on the paticula geomet of inteest. Below these foms ae povided in ectangula clindical pola and spheical pola coodinates. Rectangula coodinates Clindical pola coodinates 2 s = 1 Spheical pola coodinates 2 s = s 2 s = 2 s 2 2 s 2 2 s s 1 2 s s sin 1.7 Laplacian of a vecto u sin s 1 2 sin The Laplacian of a vecto is a vecto quantit. The foms of the diffeential Laplacian opeato depend on the paticula geomet of inteest. Below these foms ae povided in ectangula clindical pola and spheical pola coodinates. Rectangula coodinates The vecto is given b u = u i u i u z i z so u is given b: 1 u 1 15 u i. 4
5 u = 2 u 2 2 u 2 2 u z 2 2 u 2 2 u 2 2 u z 2 2 u 2 i 2 u 2 2 u z 2 i i z. 19 Clindical pola coodinates The vecto is given b u = u i u i u z i z so u is given b: u = [ ] 1 u [ ] 1 u u u u z u z 2 2 u 2 2 u 2 2 u 2 2 u 2 i i i z. 20 Spheical pola coodinates The vecto is given b u = u i u i u i so u is given b: u = u 1 2 sin u sin 2 2 u 2 u 2 2cot 2 u 2 u 2 i sin u 1 2 sin u sin 2 u sin 2 u 2 cos u 2 sin 2 i u 1 2 sin u sin 2 u 2 sin 2 cos u 2 sin sin sin 2 2 u 2 i 2 u 2 i 1 2 u 2 sin sin 2 u i i. 21 5
6 1.8 Vecto identities The elationships below ae valid fo an coodinate sstem and fo an scala s vecto fields u u 1 and u 2 and tenso T. s = 0 22 u = 0 23 u = u u 24 su = s u s u 25 st = s T s T 26 T u = T : u u T 27 si : u = s u 28 u 1 u 2 = u 1 u 2 u 2 u 1 29 u 1 u 2 = u 2 u 1 u 1 u 2 u 2 u 1 u 1 u Gauss divegence theoem The following elationships ae valid fo an vecto u and an tenso T. udv = n uds 31 V V TdV = S S n TdS 32 whee n is the oute unit nomal to an element ds of the suface S of a volume V. 6
7 2 Mass consevation continuit equations fo an incompessible fluid 2.1 Vecto fom whee u is the velocit vecto. 2.2 Catesian coodinates u = 0 33 Fo velocit vecto u = u i u i u z i z u is given b u u 2.3 Clindical pola coodinates Fo velocit vecto u = u i u i u z i z u is given b 1 u 1 u 2.4 Spheical pola coodinates Fo velocit vecto u = u i u i u i u is given b 1 2 u 2 1 u sin 1 u sin sin = = =
8 3 Momentum consevation equations fo an incompessible fluid 3.1 Vecto fom u t u u = g σ 37 whee t is time u is the velocit vecto σ is the stess tenso is the densit and g is the gavitational acceleation vecto. 3.2 Catesian coodinates Fo the following quantities: Velocit vecto u = u i u i u z i z ; Gavitational acceleation vecto g = g i g i g z i z ; Total stess tenso σ = σ i i σ i i σ z i i z σ i i σ i i σ z i i z σ z i z i σ z i z i σ zz i z i z the momentum consevation equations ae given b u t u u u u u z u t u u u u u z uz t u u u z u = g σ σ σ z 38 u = g σ σ σ z 39 = g z σ z σ z σ zz Clindical pola coodinates Fo the following quantities: Velocit vecto u = u i u i u z i z ; Gavitational acceleation vecto g = g i g i g z i z ; Total stess tenso σ = σ i i σ i i σ z i i z σ i i σ i i σ z i i z σ z i z i σ z i z i σ zz i z i z 8
9 the momentum consevation equations ae given b u t u u 1 u u 1 u 2 u u z = g 1 u t u u 1 u u 1 u u u u z uz t u 1 u u z = g z 1 σ 1 σ = g 1 2 σ 2 1 σ σ σ z 41 σ z 42 σ z 1 σ z σ zz Spheical pola coodinates Fo the following quantities: Velocit vecto u = u i u i u i ; Gavitational acceleation vecto g = g i g i g i ; Total stess tenso σ = σ i i σ i i σ i i σ i i σ i i σ i i σ i i σ i i σ i i the momentum consevation equations ae given b u t u u 1 u u 1 sin u u φ φ σ 1 sin σ sin 1 sin u 2 u φ 2 = g σ φ φ σ σ φφ 44 u t u u 1 u u 1 sin u u φ φ 1 u u cot u φ 2 = g σ 1 σ sin 1 σ φ sin sin φ σ cot σ φφ 45 uφ t u u φ 1 u u φ 1 sin u u φ φ φ 1 u φu cot u u φ = g φ σ φ 1 σ φ 1 σ φφ sin φ σ φ 2 cot σ φ. 46 9
10 4 Stess ate of stain voticit and the constitutive elation fo Newtonian fluids 4.1 Rate of stain The ate of stain tenso e fo an incompessible fluid is given b e = 1 2 Note that e is a smmetic tenso Catesian coodinates u u T. 47 The Catesian components of e fo velocit components u u u z in z diections ae e e e z e = e e e z. 48 e z e z e zz These component ae given b e = e = 1 2 e = u u u e z = e z = 1 2 e = u e z = e z = 1 2 e zz = 49 u 50 u u z. 51 Note that e = e e z = e z and e z = e z since e is smmetic Clindical pola coodinates The e components fo velocit components u u u z in diections z ae These component ae given b e z = e z = 1 2 e = u e = 1 u e = e = 1 2 e e e z e e e z e z e z e zz u u e = 1 e z = e z = 1 2 u e zz = 53 u u z u Note that e z = e z e = e and e z = e z since e is smmetic. 10
11 4.1.3 Spheical pola coodinates The e components fo velocit components u u u in diections ae These component ae given b e = u e = e = 1 2 e = 1 e = e e e e e e e e e u u e = 1 sin 1 u u u e = e = 1 2 e = e = u u u 1 u sin u cot 57 u 58 1 u 1 u sin u cot. 59 Note that e = e e = e and e = e since e is smmetic. 4.2 Stess The total stess σ is given b whee p is pessue I = stess tenso σ = pi τ 60 is the unit mati and τ is the deviatoic 4.3 Constitutive elation Fo Newtonian incompessible fluids the constitutive elation fo τ is given b the following vecto fom epession τ = 2µe 61 whee µ is the dnamic viscosit and e is the ate of stain tenso see above. 4.4 Components of stess Catesian coodinates The Catesian components of σ ae σ = σ σ σ z σ σ σ z σ z σ z σ zz
12 These components ae given b σ = p τ = p 2µ u σ = p τ = p 2µ u σ zz = p τ zz = p 2µ u σ = σ = τ = τ = µ u u σ z = σ z = τ z = τ z = µ u z uz σ z = σ z = τ z = τ z = µ u Clindical pola coodinates The components of σ in clindical pola coodinates ae σ σ σ z σ = σ σ σ z. 64 σ z σ z σ zz These components ae σ = p τ = p 2µ u 1 σ = p τ = p 2µ u u σ zz = p τ zz = p 2µ σ = σ = τ = τ = µ u 1 u u σ z = σ z = τ z = τ z = µ 1 uz σ z = σ z = τ z = τ z = µ u Spheical pola coodinates. 65 The components of σ in spheical pola coodinates ae σ σ σ σ = σ σ σ. 66 σ σ σ 12
13 These components ae σ = p τ = p 2µ u 1 u σ = p τ = p 2µ u 1 u σ = p τ = p 2µ sin u u cot σ = σ = τ = τ = µ u 1 u sin u σ = σ = τ = τ = µ sin 1 u sin 1 u σ = σ = τ = τ = µ sin u
14 4.5 Voticit Vecto fom The voticit is given b the following vecto fom epession Catesian coodinates ω = u. 68 The ω = ω i ω i ω z i z components in catesian coodinates fo a velocit vecto u = u i u i u z i z ae uz ω = u u ω = u z u ω z = u Clindical pola coodinates The ω = ω i ω i ω z i z components in clindical pola coodinates fo a velocit vecto u = u i u i u z i z ae ω = 1 u ω = u ω z = 1 u 1 u Spheical pola coodinates The ω = ω i ω i ω i components in spheical pola coodinates fo a velocit vecto u = u i u i u i ae ω = 1 sin u sin u ω = 1 u sin 1 u Rotation tenso ω = 1 u 1 u. 72 It is also useful to define the otation tenso Γ which has the following vecto fom epession Γ = 1 u u T It is possible to elate the Γ components to the ω components. We illustate this in Catesian coodinates Γ = Γ = 1 2 ω z Γ z = Γ z = 1 2 ω Γ z = Γ z = 1 2 ω. 74 Note that Γ is not smmetic unlike the ate of stain tenso e. 14
15 5 The Navie-Stokes equations fo an incompessible Newtonian fluid 5.1 Vecto fom u t u u = p g µ 2 u Catesian coodinates u t u u u u u u z u t u u u u u z uz t u u u z 5.3 Clindical coodinates = g p 2 µ u 2 2 u 2 2 u 2 u = g p 2 µ u 2 2 u 2 2 u 2 = g z p 2 µ u z 2 2 u z 2 2 u z u t u u 1 u u 1 u 2 u u z = g p 1 µ u 1 2 u u 2 2 u 2 79 u t u u 1 u u 1 u u u u z = g 1 1 µ u 1 2 u u 2 2 u 2 p 80 uz t u 1 u 1 µ u z 1 2 u z u z 2 = g z p
16 5.4 Spheical coodinates u t u u 1 u u 1 sin u u u u = g p 1 µ 2 2 u 1 2 sin u sin 1 2 u µ 2 sin u 2 2 cot 2 u 2 u 2 82 sin u t u u 1 u 1 µ 2 2 u 1 2 u µ 2 sin 2 2 u sin sin u u 1 u u cot sin u 2 2 u 1 2 sin 2 u 2 cos 2 sin 2 u 2 = g 1 p u 83 u t u u 1 u u = g 1 sin µ 1 2 sin 2 p µ 1 sin u u 1 u u cot u u u 1 2 sin u sin 2 u sin 2 u 2 u 2 sin 2 cos 2 sin 2 u
17 6 Renolds-aveaged Navie-Stokes equations 6.1 Catesian coodinates u u u t u u u u u z u u u t u u u u u z u u uz t u u u z u z u u = g p u u u u z u = g p u u u u z = g z p u z u u z u z = µ u 2 2 u 2 2 u µ u 2 2 u 2 2 u µ u z 2 2 u z 2 2 u z
18 6.2 Clindical coodinates 1 u 1 u = 0 89 u t u u 1 u u 1 u 2 u u z = g p 1 µ u 1 2 u u 2 2 u 2 u u u u u z u 1 u u 90 u t u u 1 u u 1 u u u u z = g 1 1 µ u 1 2 u u 2 2 u 2 u u u u u zu p 2 u u 91 uz t u 1 u 1 µ u u z u z 1 2 u z 2 2 u u z = g z p 2 u z 2 u z u z
19 6.3 Spheical coodinates u sin sin u 1 u sin = 0 93 u t u u 1 u u 1 sin u u u u = g p 1 µ 2 2 u 1 2 sin u sin 1 2 u µ 2 sin u 2 2 cot 2 u 2 u 2 sin 2 2 u u sin u sin u u sin u 1 u u 1 u u 94 u t u u 1 u u 1 sin u u 1 u u cot 1 µ 2 2 u 1 2 sin u sin 1 2 u µ 2 sin u sin 2 u 2 cos u 2 sin u u sin u sin u u sin u u 2 = g 1 p cot u u 95 u t u u 1 u u = g 1 sin 1 µ 3 sin p µ 2 u 2 sin sin 2 u 3 u u sin 1 sin u u 1 u u cot u u u 1 2 sin u sin 2 u 2 sin sin u u 2 cos 2 sin 2 u u u cot u u
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