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Implementation of Boundary Conditions

72

73 Momentum equations

For u -momentum Eq. 9.1.1 becomes:

π‘Žπ‘ƒπ‘’π‘ƒ = π‘Žπ‘Šπ‘’π‘Š+ π‘ŽπΈπ‘’πΈ+ π‘Žπ‘π‘’π‘+ π‘Žπ‘†π‘’π‘†+ π‘Žπ‘‡π‘’π‘‡+ π‘Žπ΅π‘’π΅+ 𝑆𝑒 (9.1.2) Where 𝑆𝑒 also includes the π‘π‘‘π‘’π‘Ÿπ‘š

At the inlet, velocities are known. In Fig. 9.1, inlet boundary is the west face of the cell with center P. The top and bottom face of the control volume can be found in the perpendicular direction of the two dimensional cell. When a Dirichlet type boundary condition is applied, for momentum equations this means, the mass flux 𝐹𝑀 is known from known velocities at inlet.

That is

𝐹𝑀 = 𝜌[𝑒𝑖𝑛𝑙𝑒𝑑{(𝑦𝑛 βˆ’ 𝑦𝑠)𝑀(π‘§π‘‘βˆ’ 𝑧𝑏)π‘€βˆ’ (π‘¦π‘‘βˆ’ 𝑦𝑏)𝑀(π‘§π‘›βˆ’ 𝑧𝑠)𝑀}

+ 𝑣𝑖𝑛𝑙𝑒𝑑{(π‘₯π‘‘βˆ’ π‘₯𝑏)𝑀(π‘§π‘›βˆ’ 𝑧𝑠)π‘€βˆ’ (π‘₯π‘›βˆ’ π‘₯𝑠)𝑀(π‘§π‘‘βˆ’ 𝑧𝑏)𝑀}

+ 𝑀𝑖𝑛𝑙𝑒𝑑{(π‘₯π‘›βˆ’ π‘₯𝑠)𝑀(π‘¦π‘‘βˆ’ 𝑦𝑏)𝑀 βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)𝑀(π‘¦π‘›βˆ’ 𝑦𝑠)𝑀}] (9.1.3) Now, Eq. 9.1.2 can be modified by using the following condition,

π‘Žπ‘€ = 0 (9.1.4)

π‘Žπ‘ = {𝐷𝑀+ π‘šπ‘Žπ‘₯(𝐹𝑀, 0)} + π‘ŽπΈ + π‘Žπ‘+ π‘Žπ‘†+ π‘Žπ‘‡+ π‘Žπ΅+ π‘Žπ‘ƒπ‘œ

+ [πΉπ‘’βˆ’ 𝐹𝑀 + πΉπ‘›βˆ’ 𝐹𝑠+ πΉπ‘‘βˆ’ 𝐹𝑏] (9.1.5)

𝑆𝑒 = [{𝑁𝑒1(π‘’π‘›βˆ’ 𝑒𝑠)𝑒+ 𝑁𝑒2(π‘’π‘‘βˆ’ 𝑒𝑏)𝑒}

βˆ’ {𝑁𝑀1(π‘’π‘›βˆ’ 𝑒𝑠)𝑀 + 𝑁𝑀2(π‘’π‘‘βˆ’ 𝑒𝑏)𝑀} + {𝑁𝑛1(π‘’π‘’βˆ’ 𝑒𝑀)𝑛 + 𝑁𝑛2(π‘’π‘‘βˆ’ 𝑒𝑏)𝑛}

βˆ’ {𝑁𝑠1(π‘’π‘’βˆ’ 𝑒𝑀)𝑠+ 𝑁𝑠2(π‘’π‘‘βˆ’ 𝑒𝑏)𝑠} + {𝑁𝑑1(π‘’π‘’βˆ’ 𝑒𝑀)𝑑+ 𝑁𝑑2(π‘’π‘›βˆ’ 𝑒𝑠)𝑑}

βˆ’ {𝑁𝑏1(π‘’π‘’βˆ’ 𝑒𝑀)𝑏+ 𝑁𝑏2(π‘’π‘›βˆ’ 𝑒𝑠)𝑏}]

+ πœ“{[πΉπ‘’π‘’π‘’βˆ’ 𝐹𝑀𝑒𝑀 + 𝐹𝑛𝑒𝑛 βˆ’ 𝐹𝑠𝑒𝑠+πΉπ‘‘π‘’π‘‘βˆ’ 𝐹𝑏𝑒𝑏]π‘ˆπ·π‘†

βˆ’ [πΉπ‘’π‘’π‘’βˆ’ 𝐹𝑀𝑒𝑀+ 𝐹𝑛𝑒𝑛 βˆ’ 𝐹𝑠𝑒𝑠+πΉπ‘‘π‘’π‘‘βˆ’ 𝐹𝑏𝑒𝑏]𝐢𝐷𝑆}

+ {𝐷𝑀+ π‘šπ‘Žπ‘₯(𝐹𝑀, 0)}𝑒𝑖𝑛𝑙𝑒𝑑+ π‘Žπ‘ƒπ‘œπ‘’π‘ƒπ‘œ+ π‘π‘‘π‘’π‘Ÿπ‘š (9.1.6)

Other quantities remain unchanged.

Turbulent kinetic energy and dissipation equations

π‘Žπ‘ƒπ‘˜π‘ƒ = π‘Žπ‘Šπ‘˜π‘Š+ π‘ŽπΈπ‘˜πΈ+ π‘Žπ‘π‘˜π‘+ π‘Žπ‘†π‘˜π‘†+ π‘Žπ‘‡π‘˜π‘‡+ π‘Žπ΅π‘˜π΅+ π‘†π‘˜ (9.1.7) π‘Žπ‘ƒΞ΅π‘ƒ = π‘Žπ‘ŠΞ΅π‘Š+ π‘ŽπΈΞ΅πΈ+ π‘Žπ‘Ξ΅π‘+ π‘Žπ‘†Ξ΅π‘†+ π‘Žπ‘‡Ξ΅π‘‡+ π‘Žπ΅Ξ΅π΅+ 𝑆Ρ (9.1.8)

74 where for turbulent kinetic energy we have extra terms embedded in π‘Žπ‘ƒ and π‘†π‘˜ arising from production and dissipation as follows:

π‘Žπ‘ƒ = π‘Žπ‘Š+ π‘ŽπΈ+ π‘Žπ‘+ π‘Žπ‘†+ π‘Žπ‘‡ + π‘Žπ΅+ π‘Žπ‘ƒπ‘œ+πœ€π‘βˆ— π‘˜π‘βˆ—π‘˜π‘ƒπ›Ώπ‘‰

+ [πΉπ‘’βˆ’ 𝐹𝑀 + πΉπ‘›βˆ’ 𝐹𝑠+ πΉπ‘‘βˆ’ 𝐹𝑏] (9.1.9)

π‘†π‘˜= [{𝑁𝑒1(π‘˜π‘› βˆ’ π‘˜π‘ )𝑒+ 𝑁𝑒2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑒}

βˆ’ {𝑁𝑀1(π‘˜π‘›βˆ’ π‘˜π‘ )𝑀+ 𝑁𝑀2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑀} + {𝑁𝑛1(π‘˜π‘’βˆ’ π‘˜π‘€)𝑛 + 𝑁𝑛2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑛}

βˆ’ {𝑁𝑠1(π‘˜π‘’βˆ’ π‘˜π‘€)𝑠+ 𝑁𝑠2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑠} + {𝑁𝑑1(π‘˜π‘’ βˆ’ π‘˜π‘€)𝑑+ 𝑁𝑑2(π‘˜π‘›βˆ’ π‘˜π‘ )𝑑}

βˆ’ {𝑁𝑏1(π‘˜π‘’βˆ’ π‘˜π‘€)𝑏+ 𝑁𝑏2(π‘˜π‘›βˆ’ π‘˜π‘ )𝑏}]

+ πœ“{[πΉπ‘’π‘˜π‘’βˆ’ πΉπ‘€π‘˜π‘€+ πΉπ‘›π‘˜π‘› βˆ’ πΉπ‘ π‘˜π‘ +πΉπ‘‘π‘˜π‘‘βˆ’ πΉπ‘π‘˜π‘]π‘ˆπ·π‘†

βˆ’ [πΉπ‘’π‘˜π‘’βˆ’ πΉπ‘€π‘˜π‘€+ πΉπ‘›π‘˜π‘› βˆ’ πΉπ‘ π‘˜π‘ +πΉπ‘‘π‘˜π‘‘βˆ’ πΉπ‘π‘˜π‘]𝐢𝐷𝑆}

+ (𝐺𝑃)𝛿𝑉 + π‘Žπ‘ƒπ‘œπ‘˜π‘ƒπ‘œ (9.1.10)

And similarly for turbulent dissipation equation we have extra terms embedded in π‘Žπ‘ƒ and π‘†π‘˜ as follows:

π‘Žπ‘ƒ = π‘Žπ‘Š+ π‘ŽπΈ+ π‘Žπ‘+ π‘Žπ‘†+ π‘Žπ‘‡ + π‘Žπ΅+ π‘Žπ‘ƒπ‘œ+πœ€π‘βˆ— π‘˜π‘ƒπœ€π‘ƒ

+ [πΉπ‘’βˆ’ 𝐹𝑀 + πΉπ‘›βˆ’ 𝐹𝑠+ πΉπ‘‘βˆ’ 𝐹𝑏] (9.1.11)

π‘†πœ€ = [{𝑁𝑒1(πœ€π‘›βˆ’ πœ€π‘ )𝑒+ 𝑁𝑒2(πœ€π‘‘βˆ’ πœ€π‘)𝑒}

βˆ’ {𝑁𝑀1(πœ€π‘› βˆ’ πœ€π‘ )𝑀 + 𝑁𝑀2(πœ€π‘‘βˆ’ πœ€π‘)𝑀} + {𝑁𝑛1(πœ€π‘’βˆ’ πœ€π‘€)𝑛+ 𝑁𝑛2(πœ€π‘‘βˆ’ πœ€π‘)𝑛}

βˆ’ {𝑁𝑠1(πœ€π‘’βˆ’ πœ€π‘€)𝑠+ 𝑁𝑠2(πœ€π‘‘βˆ’ πœ€π‘)𝑠} + {𝑁𝑑1(πœ€π‘’βˆ’ πœ€π‘€)𝑑+ 𝑁𝑑2(πœ€π‘›βˆ’ πœ€π‘ )𝑑}

βˆ’ {𝑁𝑏1(πœ€π‘’βˆ’ πœ€π‘€)𝑏+ 𝑁𝑏2(πœ€π‘›βˆ’ πœ€π‘ )𝑏}]

+πœ“{[πΉπ‘’πœ€π‘’βˆ’ πΉπ‘€πœ€π‘€+ πΉπ‘›πœ€π‘›βˆ’ πΉπ‘ πœ€π‘ +πΉπ‘‘πœ€π‘‘βˆ’ πΉπ‘πœ€π‘]π‘ˆπ·π‘†

βˆ’ [πΉπ‘’πœ€π‘’βˆ’ πΉπ‘€πœ€π‘€+ πΉπ‘›πœ€π‘›βˆ’ πΉπ‘ πœ€π‘ +πΉπ‘‘πœ€π‘‘βˆ’ πΉπ‘πœ€π‘]𝐢𝐷𝑆} + πΆπœ€1πœ€π‘βˆ—

π‘˜π‘ƒπΊπ‘ƒπ›Ώπ‘‰ + π‘Žπ‘ƒπ‘œπœ€π‘ƒπ‘œ

(9.1.12) With mass flux given by prescribed velocities, one needs to provide known values at inlet for turbulent kinetic energy and dissipation. That is, π›·π‘Š = π‘˜π‘€ = π‘˜π‘–π‘›π‘™π‘’π‘‘ and, π›·π‘Š = πœ€π‘€ = πœ€π‘–π‘›π‘™π‘’π‘‘ . For Turbulent kinetic energy this means

75

π‘Žπ‘€ = 0 (9.1.13)

π‘Žπ‘ƒ = {𝐷𝑀 + π‘šπ‘Žπ‘₯(𝐹𝑀, 0)} + π‘ŽπΈ+ π‘Žπ‘+ π‘Žπ‘†+ π‘Žπ‘‡+ π‘Žπ΅+ π‘Žπ‘ƒπ‘œ+ πœ€π‘βˆ— π‘˜π‘βˆ—π‘˜π‘ƒπ›Ώπ‘‰

+ [πΉπ‘’βˆ’ 𝐹𝑀 + πΉπ‘›βˆ’ 𝐹𝑠+ πΉπ‘‘βˆ’ 𝐹𝑏] (9.1.14)

π‘†π‘˜= [{𝑁𝑒1(π‘˜π‘› βˆ’ π‘˜π‘ )𝑒+ 𝑁𝑒2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑒}

βˆ’ {𝑁𝑀1(π‘˜π‘›βˆ’ π‘˜π‘ )𝑀+ 𝑁𝑀2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑀} + {𝑁𝑛1(π‘˜π‘’βˆ’ π‘˜π‘€)𝑛 + 𝑁𝑛2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑛}

βˆ’ {𝑁𝑠1(π‘˜π‘’βˆ’ π‘˜π‘€)𝑠+ 𝑁𝑠2(π‘˜π‘‘βˆ’ π‘˜π‘)𝑠} + {𝑁𝑑1(π‘˜π‘’ βˆ’ π‘˜π‘€)𝑑+ 𝑁𝑑2(π‘˜π‘›βˆ’ π‘˜π‘ )𝑑}

βˆ’ {𝑁𝑏1(π‘˜π‘’βˆ’ π‘˜π‘€)𝑏+ 𝑁𝑏2(π‘˜π‘›βˆ’ π‘˜π‘ )𝑏}]

+ πœ“{[πΉπ‘’π‘˜π‘’βˆ’ πΉπ‘€π‘˜π‘€+ πΉπ‘›π‘˜π‘› βˆ’ πΉπ‘ π‘˜π‘ +πΉπ‘‘π‘˜π‘‘βˆ’ πΉπ‘π‘˜π‘]π‘ˆπ·π‘†

βˆ’ [πΉπ‘’π‘˜π‘’βˆ’ πΉπ‘€π‘˜π‘€+ πΉπ‘›π‘˜π‘› βˆ’ πΉπ‘ π‘˜π‘ +πΉπ‘‘π‘˜π‘‘βˆ’ πΉπ‘π‘˜π‘]𝐢𝐷𝑆}

+ {𝐷𝑀+ π‘šπ‘Žπ‘₯(𝐹𝑀, 0)}π‘˜π‘–π‘›π‘™π‘’π‘‘+ (𝐺𝑃)𝛿𝑉 + π‘Žπ‘ƒπ‘œπ‘˜π‘ƒπ‘œ (9.1.15) For Turbulent dissipation equation this means,

π‘Žπ‘€ = 0 (9.1.16)

π‘Žπ‘ƒ = {𝐷𝑀 + π‘šπ‘Žπ‘₯(𝐹𝑀, 0)} + π‘ŽπΈ+ π‘Žπ‘+ π‘Žπ‘†+ π‘Žπ‘‡+ π‘Žπ΅+ π‘Žπ‘ƒπ‘œ+ πœ€π‘βˆ— π‘˜π‘ƒπœ€π‘ƒ

+ [πΉπ‘’βˆ’ 𝐹𝑀 + πΉπ‘›βˆ’ 𝐹𝑠+ πΉπ‘‘βˆ’ 𝐹𝑏] (9.1.17)

π‘†πœ€ = [{𝑁𝑒1(πœ€π‘›βˆ’ πœ€π‘ )𝑒+ 𝑁𝑒2(πœ€π‘‘βˆ’ πœ€π‘)𝑒}

βˆ’ {𝑁𝑀1(πœ€π‘› βˆ’ πœ€π‘ )𝑀 + 𝑁𝑀2(πœ€π‘‘βˆ’ πœ€π‘)𝑀} + {𝑁𝑛1(πœ€π‘’βˆ’ πœ€π‘€)𝑛+ 𝑁𝑛2(πœ€π‘‘βˆ’ πœ€π‘)𝑛}

βˆ’ {𝑁𝑠1(πœ€π‘’βˆ’ πœ€π‘€)𝑠+ 𝑁𝑠2(πœ€π‘‘βˆ’ πœ€π‘)𝑠} + {𝑁𝑑1(πœ€π‘’βˆ’ πœ€π‘€)𝑑+ 𝑁𝑑2(πœ€π‘›βˆ’ πœ€π‘ )𝑑}

βˆ’ {𝑁𝑏1(πœ€π‘’βˆ’ πœ€π‘€)𝑏+ 𝑁𝑏2(πœ€π‘›βˆ’ πœ€π‘ )𝑏}]

+ πœ“{[πΉπ‘’πœ€π‘’βˆ’ πΉπ‘€πœ€π‘€+ πΉπ‘›πœ€π‘›βˆ’ πΉπ‘ πœ€π‘ +πΉπ‘‘πœ€π‘‘βˆ’ πΉπ‘πœ€π‘]π‘ˆπ·π‘†

βˆ’ [πΉπ‘’πœ€π‘’βˆ’ πΉπ‘€πœ€π‘€+ πΉπ‘›πœ€π‘›βˆ’ πΉπ‘ πœ€π‘ +πΉπ‘‘πœ€π‘‘βˆ’ πΉπ‘πœ€π‘]𝐢𝐷𝑆} + {𝐷𝑀+ π‘šπ‘Žπ‘₯(𝐹𝑀, 0)}πœ€π‘–π‘›π‘™π‘’π‘‘+ πΆπœ€1πœ€π‘βˆ—

π‘˜π‘ƒπΊπ‘ƒπ›Ώπ‘‰ + π‘Žπ‘ƒπ‘œπœ€π‘ƒπ‘œ

(9.1.18)

Pressure correction equations

Since the inlet boundary mass fluxes are known (or treated as such), they need not be corrected.

This is equivalent to prescribing zero-gradient boundary conditions for the pressure-correction equation, which is readily implemented by setting to zero the coefficient (π‘Žπ‘Š in this case) of

76 pressure correction equation, corresponding to the boundary node. The same treatment also applies for outlet, symmetry and wall boundaries. Pressure correction equation for a control volume near the west boundary is given below.

π‘Žπ‘ƒπ‘π‘ƒβ€² = π‘Žπ‘Šπ‘π‘Šβ€² + π‘ŽπΈπ‘πΈβ€² + π‘Žπ‘π‘π‘β€² + π‘Žπ‘†π‘π‘†β€² + π‘Žπ‘‡π‘π‘‡β€² + π‘Žπ΅π‘π΅β€² + b (9.1.19) Where

π‘Žπ‘ƒ = π‘Žπ‘Š+ π‘ŽπΈ+ π‘Žπ‘+ π‘Žπ‘†+ π‘Žπ‘‡ + π‘Žπ΅ (9.1.20)

π‘Žπ‘Š = 0 (9.1.21)

π‘ŽπΈ = 𝜌 { 𝛼𝑒( 1

π‘Žπ‘ƒ)𝑒{(π‘¦π‘›βˆ’ 𝑦𝑠)(π‘§π‘‘βˆ’ 𝑧𝑏) βˆ’ (π‘¦π‘‘βˆ’ 𝑦𝑏)(π‘§π‘›βˆ’ 𝑧𝑠)}𝑒2 + 𝛼𝑣( 1

π‘Žπ‘ƒ)𝑒{(π‘₯π‘‘βˆ’ π‘₯𝑏)(π‘§π‘›βˆ’ 𝑧𝑠) βˆ’ (π‘₯π‘›βˆ’ π‘₯𝑠)(π‘§π‘‘βˆ’ 𝑧𝑏)}𝑒2 + 𝛼𝑀( 1

π‘Žπ‘ƒ)𝑒{(π‘₯π‘›βˆ’ π‘₯𝑠)(π‘¦π‘‘βˆ’ 𝑦𝑏) βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)(π‘¦π‘›βˆ’ 𝑦𝑠)}𝑒2}

(9.1.22)

π‘Žπ‘ = 𝜌 { 𝛼𝑒( 1

π‘Žπ‘ƒ)𝑛{(π‘¦π‘‘βˆ’ 𝑦𝑏)(π‘§π‘’βˆ’ 𝑧𝑀) βˆ’ (π‘¦π‘’βˆ’ 𝑦𝑀)(π‘§π‘‘βˆ’ 𝑧𝑏)}𝑛2 + 𝛼𝑣( 1

π‘Žπ‘ƒ)𝑛{(π‘₯π‘’βˆ’ π‘₯𝑀)(π‘§π‘‘βˆ’ 𝑧𝑏) βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)(π‘§π‘’βˆ’ 𝑧𝑀)}𝑛2 + 𝛼𝑀( 1

π‘Žπ‘ƒ)𝑛{(π‘₯π‘‘βˆ’ π‘₯𝑏)(π‘¦π‘’βˆ’ 𝑦𝑀) βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)(π‘¦π‘‘βˆ’ 𝑦𝑏)}𝑛2}

(9.1.23)

π‘Žπ‘† = 𝜌 { 𝛼𝑒( 1

π‘Žπ‘ƒ)𝑠{(π‘¦π‘‘βˆ’ 𝑦𝑏)(π‘§π‘’βˆ’ 𝑧𝑀) βˆ’ (π‘¦π‘’βˆ’ 𝑦𝑀)(π‘§π‘‘βˆ’ 𝑧𝑏)}𝑠2 + 𝛼𝑣( 1

π‘Žπ‘ƒ)𝑠{(π‘₯π‘’βˆ’ π‘₯𝑀)(π‘§π‘‘βˆ’ 𝑧𝑏) βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)(π‘§π‘’βˆ’ 𝑧𝑀)}𝑠2 + 𝛼𝑀( 1

π‘Žπ‘ƒ)𝑠{(π‘₯π‘‘βˆ’ π‘₯𝑏)(π‘¦π‘’βˆ’ 𝑦𝑀) βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)(π‘¦π‘‘βˆ’ 𝑦𝑏)}𝑠2}

(9.1.24)

π‘Žπ‘‡ = 𝜌 { 𝛼𝑒( 1

π‘Žπ‘ƒ)𝑑{(𝑦𝑒 βˆ’ 𝑦𝑀)(π‘§π‘›βˆ’ 𝑧𝑠) βˆ’ (π‘¦π‘›βˆ’ 𝑦𝑠)(π‘§π‘’βˆ’ 𝑧𝑀)}𝑑2 + 𝛼𝑣( 1

π‘Žπ‘ƒ)𝑑{(π‘₯π‘›βˆ’ π‘₯𝑠)(π‘§π‘’βˆ’ 𝑧𝑀) βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)(π‘§π‘›βˆ’ 𝑧𝑠)}𝑑2 + 𝛼𝑀( 1

π‘Žπ‘ƒ)𝑑{(π‘₯π‘’βˆ’ π‘₯𝑀)(π‘¦π‘›βˆ’ 𝑦𝑠) βˆ’ (π‘₯π‘›βˆ’ π‘₯𝑠)(π‘¦π‘’βˆ’ 𝑦𝑀)}𝑑2}

(9.1.25)

77 π‘Žπ΅ = 𝜌 { 𝛼𝑒( 1

π‘Žπ‘ƒ)𝑏{(π‘¦π‘’βˆ’ 𝑦𝑀)(π‘§π‘›βˆ’ 𝑧𝑠) βˆ’ (𝑦𝑛 βˆ’ 𝑦𝑠)(π‘§π‘’βˆ’ 𝑧𝑀)}𝑏2 + 𝛼𝑣( 1

π‘Žπ‘ƒ)𝑏{(π‘₯π‘›βˆ’ π‘₯𝑠)(π‘§π‘’βˆ’ 𝑧𝑀) βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)(π‘§π‘›βˆ’ 𝑧𝑠)}𝑏2 + 𝛼𝑀( 1

π‘Žπ‘ƒ)𝑏{(π‘₯π‘’βˆ’ π‘₯𝑀)(π‘¦π‘›βˆ’ 𝑦𝑠) βˆ’ (π‘₯π‘›βˆ’ π‘₯𝑠)(π‘¦π‘’βˆ’ 𝑦𝑀)}𝑏2}

(9.1.26) 𝑏 = 𝜌[𝑒𝑖𝑛𝑙𝑒𝑑{(π‘¦π‘›βˆ’ 𝑦𝑠)𝑀(π‘§π‘‘βˆ’ 𝑧𝑏)𝑀 βˆ’ (π‘¦π‘‘βˆ’ 𝑦𝑏)𝑀(π‘§π‘›βˆ’ 𝑧𝑠)𝑀}

+ 𝑣𝑖𝑛𝑙𝑒𝑑{(π‘₯π‘‘βˆ’ π‘₯𝑏)𝑀(π‘§π‘›βˆ’ 𝑧𝑠)π‘€βˆ’ (π‘₯π‘›βˆ’ π‘₯𝑠)𝑀(π‘§π‘‘βˆ’ 𝑧𝑏)𝑀} + 𝑀𝑖𝑛𝑙𝑒𝑑{(π‘₯π‘›βˆ’ π‘₯𝑠)𝑀(π‘¦π‘‘βˆ’ 𝑦𝑏)𝑀 βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)𝑀(π‘¦π‘›βˆ’ 𝑦𝑠)𝑀}]

βˆ’ 𝜌[π‘’π‘’βˆ—{(π‘¦π‘›βˆ’ 𝑦𝑠)𝑒(π‘§π‘‘βˆ’ 𝑧𝑏)π‘’βˆ’ (π‘¦π‘‘βˆ’ 𝑦𝑏)𝑒(π‘§π‘›βˆ’ 𝑧𝑠)𝑒} + π‘£π‘’βˆ—{(π‘₯π‘‘βˆ’ π‘₯𝑏)𝑒(π‘§π‘›βˆ’ 𝑧𝑠)π‘’βˆ’ (π‘₯π‘›βˆ’ π‘₯𝑠)𝑒(π‘§π‘‘βˆ’ 𝑧𝑏)𝑒} + π‘€π‘’βˆ—{(π‘₯π‘›βˆ’ π‘₯𝑠)𝑒(π‘¦π‘‘βˆ’ 𝑦𝑏)π‘’βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)𝑒(π‘¦π‘›βˆ’ 𝑦𝑠)𝑒}]

+ 𝜌[π‘’π‘ βˆ—{(π‘¦π‘‘βˆ’ 𝑦𝑏)𝑠(π‘§π‘’βˆ’ 𝑧𝑀)π‘ βˆ’ (π‘¦π‘’βˆ’ 𝑦𝑀)𝑠(π‘§π‘‘βˆ’ 𝑧𝑏)𝑠} + π‘£π‘ βˆ—{(π‘₯π‘’βˆ’ π‘₯𝑀)𝑠(π‘§π‘‘βˆ’ 𝑧𝑏)π‘ βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)𝑠(π‘§π‘’βˆ’ 𝑧𝑀)𝑠} + π‘€π‘ βˆ—{(π‘₯π‘‘βˆ’ π‘₯𝑏)𝑠(π‘¦π‘’βˆ’ 𝑦𝑀)π‘ βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)𝑠(π‘¦π‘‘βˆ’ 𝑦𝑏)𝑠}]

βˆ’ 𝜌[π‘’π‘›βˆ—{(π‘¦π‘‘βˆ’ 𝑦𝑏)𝑛(π‘§π‘’βˆ’ 𝑧𝑀)𝑛 βˆ’ (π‘¦π‘’βˆ’ 𝑦𝑀)𝑛(π‘§π‘‘βˆ’ 𝑧𝑏)𝑛} + π‘£π‘›βˆ—{(π‘₯π‘’βˆ’ π‘₯𝑀)𝑛(π‘§π‘‘βˆ’ 𝑧𝑏)π‘›βˆ’ (π‘₯π‘‘βˆ’ π‘₯𝑏)𝑛(π‘§π‘’βˆ’ 𝑧𝑀)𝑛} + π‘€π‘›βˆ—{(π‘₯π‘‘βˆ’ π‘₯𝑏)𝑛(π‘¦π‘’βˆ’ 𝑦𝑀)π‘›βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)𝑛(π‘¦π‘‘βˆ’ 𝑦𝑏)𝑛}]

+ 𝜌[π‘’π‘βˆ—{(π‘¦π‘’βˆ’ 𝑦𝑀)𝑏(π‘§π‘›βˆ’ 𝑧𝑠)π‘βˆ’ (π‘¦π‘›βˆ’ 𝑦𝑠)𝑏(π‘§π‘’βˆ’ 𝑧𝑀)𝑏} + π‘£π‘βˆ—{(π‘₯π‘›βˆ’ π‘₯𝑠)𝑏(π‘§π‘’βˆ’ 𝑧𝑀)π‘βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)𝑏(π‘§π‘›βˆ’ 𝑧𝑠)𝑏} + π‘€π‘βˆ—{(π‘₯π‘’βˆ’ π‘₯𝑀)𝑏(π‘¦π‘›βˆ’ 𝑦𝑠)π‘βˆ’ (π‘₯𝑛 βˆ’ π‘₯𝑠)𝑏(π‘¦π‘’βˆ’ 𝑦𝑀)𝑏}]

βˆ’ 𝜌[π‘’π‘‘βˆ—{(π‘¦π‘’βˆ’ 𝑦𝑀)𝑑(𝑧𝑛 βˆ’ 𝑧𝑠)π‘‘βˆ’ (π‘¦π‘›βˆ’ 𝑦𝑠)𝑑(π‘§π‘’βˆ’ 𝑧𝑀)𝑑} + π‘£π‘‘βˆ—{(π‘₯𝑛 βˆ’ π‘₯𝑠)𝑑(π‘§π‘’βˆ’ 𝑧𝑀)π‘‘βˆ’ (π‘₯π‘’βˆ’ π‘₯𝑀)𝑑(π‘§π‘›βˆ’ 𝑧𝑠)𝑑}

+ π‘€π‘‘βˆ—{(π‘₯π‘’βˆ’ π‘₯𝑀)𝑑(𝑦𝑛 βˆ’ 𝑦𝑠)π‘‘βˆ’ (π‘₯π‘›βˆ’ π‘₯𝑠)𝑑(π‘¦π‘’βˆ’ 𝑦𝑀)𝑑}] (9.1.27)

Outlet boundary

If the values of the dependent variables are known at exit boundaries, these are then fixed as such. However, it is rare that such information is known in advance. In such cases the outlet boundary should be placed far downstream from the region of interest, at a location where the flow is everywhere directed outwards so that any inaccuracy in estimating the outlet conditions will not propagate far upstream, provided that the Reynolds number is large. In Fig. 9.2, outlet boundary is shown for a two-dimensional cell. The top and bottom face of the control volume

78 can be found in the perpendicular direction of the two-dimensional cell. If the outlet boundary is situated far away from the region of interest, then a Neumann type boundary condition can be applied.

Fig. 9.2: Cell adjacent to outlet boundary (two dimensional) Mathematically this implies:

πœ•π›·

πœ•π‘› = 0 (9.1.28)

Momentum equations

Using Neumann type boundary condition for momentum equation we can write

πœ•π‘’

πœ•π‘›,πœ•π‘£

πœ•π‘›,πœ•π‘€

πœ•π‘› , = 0 (9.1.29)

For the outlet boundary shown in Fig. 9.2 this can be implemented simply by setting

𝑒𝑃 = 𝑒𝑒, 𝑣𝑃 = 𝑣𝑒, 𝑀𝑃 = 𝑀𝑒 (9.1.30)

The modification of the algebraic form the equations may be carried out in a similar fashion as shown earlier.

In the case of the velocity components, it is also necessary to ensure that they satisfy overall continuity, i.e. that sum of the outlet mass fluxes πΉπ‘œ must equal sum of inlet mass fluxes 𝐹𝐼 .This can be done by calculating outlet mass fluxes using newly estimated values of velocity components and correcting these components by multiplying them with the ratio πΉπΉπ‘œπΌ.

Turbulent kinetic energy and dissipation equation Using Neumann type boundary condition we can write,

79

βˆ‚π‘˜

βˆ‚π‘›,βˆ‚Ο΅

βˆ‚π‘› = 0 (9.1.31)

Hence,

π‘˜π‘ƒ = π‘˜π‘’, πœ€π‘ƒ = πœ€π‘’ (9.1.32)

The modification of the algebraic form the equations may be carried out in a similar fashion as shown earlier

Pressure correction equations

Once velocities are known at the outlet boundary, pressure correction equation will be treated just like inlet boundary.

Symmetry boundary

The conditions applying at an axis or plane of symmetry are:

- no cross-flow (zero convective flux);

- zero diffusion flux of dependent variables in the normal direction.

For a boundary aligned with the xy-axis as shown in Fig. 9.3, zero convective flux can be implemented by setting the normal velocity at the boundary to be zero that is 𝑀 = 0. The values of the dependent variables at symmetry boundary nodes are needed for evaluation of fluxes through cell faces which intersect this boundary (for the situation shown in Fig. 8.3, these are e, w, n and s cell faces). These values can be calculated from the interior nodal values by imposing the condition of zero normal gradient (which is equivalent to specifying the zero- diffusion flux), i.e.:

πœ•π›·

πœ•π‘› = 0 (9.1.33)

Here n denotes the coordinate normal to the symmetry plane or axis. So,

𝛷𝑏 = 𝛷𝑃 (9.1.34)

Where

𝛷 = 𝑒, 𝑣, π‘˜, Ξ΅ (9.1.35)

80 Fig. 9.3: Cell adjacent to symmetry boundary

For a boundary aligned with the xz-axis, the symmetry boundary condition is as follows,

𝑣 = 0, 𝛷𝑛 = 𝛷𝑃 (9.1.36)

Where,

𝛷 = 𝑒, 𝑀, π‘˜, Ξ΅ (9.1.37)

Wall boundary

Implementation of Wall functions

Rigid impermeable walls occur in most fluid flow problems. The values of the velocity components (u, v, w) and hence the mass fluxes 𝐹𝑒, 𝐹𝑀, 𝐹𝑛, 𝐹𝑠, 𝐹𝑑, 𝐹𝑏 are known to be zero there.

These conditions are easy to impose; however, in the case of turbulent flows the calculation of the stresses on the wall needs special treatment, since expression (7.2.33) becomes inappropriate for the wall cell faces (Launder and Spalding, 1974; Gosman and Ideriah, 1976).

This is due to existence of boundary layers, across which steep variations of flow properties occur and the standard π‘˜ βˆ’ πœ€ model of turbulence becomes inadequate (Launder and Spalding, 1974). In order to adequately avoid these problems it would be necessary to employ a fine grid and use a low Reynolds number form of the π‘˜ βˆ’ πœ€ model of turbulence (Jones and Launder, 1972), which would be expensive. An alternative and widely employed approach is to use formulae known as "wall functions" (Launder and Spalding, 1974) to bridge this region: this has been followed in the present study. It is based on a one-dimensional Couette flow analysis,

z

y x

81 which is assumed to be valid in the near-wall region, and will be summarized briefly in what follows.

Fig. 9.4: Wall boundary

The wall shear stress 𝜏 𝑀 is expressed as a function of the adjacent nodal velocity component parallel to the wall, 𝑉⃗ 𝑝 , thus (see Fig. 9.4):

𝜏 𝑀 = βˆ’πœ†π‘€ (𝑉⃗ 𝑝)𝑃 (9.1.38)

where the coefficient πœ†π‘€ is determined from a two-part "universal" velocity profile expression (Gosman and Ideriah, 1976), ie.:

a) laminar sublayer:

πœ†π‘€ = πœ‡

𝛿𝑛 if 𝑦𝑃+ < 11.6 (9.1.39)

b) fully turbulent layer:

πœ†π‘€ = πœŒπΆπœ‡1/4π‘˜π‘ƒ1/2ΞΊ

ln (𝐸𝑦𝑃+) if 𝑦𝑃+ > 11.6 (9.1.40) Where 𝑦𝑃+ is calculated as:

𝑦𝑃+ = πœŒπΆπœ‡1/4π‘˜π‘ƒ1/2𝛿𝑛

ΞΌ (9.1.41)

x y

z

82 In the above equations ΞΊ is the Von Karman's constant (πœ… = 0.4187) and E is an integration constant which depends on the wall roughness, and through which other effects such as pressure gradients and mass transfer can be accounted for (Launder and Spalding, 1974). For smooth impermeable walls E is assigned a constant value of 9.0 (Launder and Spalding, 1974).

Since the momentum equations are resolved in the directions of the Cartesian coordinates, the resultant shear wall force 𝑇⃗ 𝑀 needs also be expressed in terms of its Cartesian components. For simplicity, it is assumed here that 𝑇⃗ 𝑀 acts in the direction opposite to 𝑉⃗ 𝑝, as shown in Fig. 9.4.

This force is equal to the product of the wall shear stress 𝜏 𝑀 and the area 𝐴𝑀 , ie.:

𝑇⃗ 𝑀 = 𝜏 𝑀𝐴𝑀 = βˆ’πœ†π‘€π΄π‘€(𝑉⃗ 𝑝)𝑃 (9.1.42)

Its components in the x, y and z directions can readily be obtained if the 𝑉⃗ 𝑝 is expressed in terms of its components in these directions, ie.:

𝑇𝑀𝑖 = βˆ’πœ†π‘€π΄π‘€(𝑉𝑝𝑖)𝑃 (9.1.43)

𝑉⃗⃗⃗ 𝑝 can be obtained by subtracting from the total velocity 𝑉⃗ its component in the direction normal to the wall, 𝑉⃗ 𝑛 , which is deducible from the scalar product of 𝑉⃗ and the unit normal vector 𝑛⃗ . If the Cartesian components of 𝑛⃗ are 𝑛1 , 𝑛2 and 𝑛3 , then the following expressions for shear force components result:

𝑇𝑀π‘₯ = βˆ’πœ†π‘€π΄π‘€{[1 βˆ’ (𝑛1)2]π‘’π‘ƒβˆ’ 𝑛1𝑛2𝑣𝑃 βˆ’ 𝑛1𝑛3𝑀𝑃} (9.1.44) 𝑇𝑀𝑦 = βˆ’πœ†π‘€π΄π‘€{βˆ’π‘›1𝑛2𝑒𝑃+ [1 βˆ’ (𝑛2)2]π‘£π‘ƒβˆ’ 𝑛2𝑛3𝑀𝑃} (9.1.45) 𝑇𝑀𝑧 = βˆ’πœ†π‘€π΄π‘€{βˆ’π‘›1𝑛3𝑒𝑃 βˆ’ 𝑛2𝑛3𝑣𝑃+ [1 βˆ’ (𝑛3)2]𝑀𝑃} (9.1.46) The normal vector 𝑛⃗ is defined by the vector product of two cell face "central lines", eg. for the situation shown in Fig. 8.4, from (𝑒𝑀⃗⃗⃗⃗⃗ )𝑏 and (𝑛𝑠⃗⃗⃗⃗ )𝑏.

Thus, for the momentum equations the shear stress integrals over boundary cell faces, appearing in equation like (7.2.33), are replaced by the 𝑇𝑀𝑖 , defined in (9.1.44-9.1.46).

The equations for k and Ξ΅ also need special treatment for the near wall cells. In the equation for k, the diffusion flux through the wall cell face is set to zero (Launder and Spalding, 1974), while the source terms are modified as follows.

Within the turbulent layer a balance between production and dissipation of turbulent kinetic energy is assumed (Launder and Spalding, 1972), giving the following relation for Ξ΅:

83 πœ€ =πΆπœ‡3/4π‘˜3/2

ΞΊ y (9.1.47)

The dissipation of k is now obtained by integration over the near-wall cell (Gosman and Ideriah, 1976):

βˆ’βˆ«π›Ώπ‘‰ πœŒπœ€π‘‘π‘‰ ≃ βˆ’π‘†π‘˜β€²β€² π‘˜π‘ƒ = βˆ’πœŒπΆπœ‡

3 4π‘˜π‘ƒβˆ—

1 2𝑒𝑃+𝛿𝑉𝑃

𝑦𝑃 π‘˜π‘ƒ (9.1.48)

where 𝑒𝑃+ is given by:

𝑒𝑃+ = 𝑦𝑃+ , if 𝑦𝑃+ < 11.6 𝑒𝑃+ = 1

ΞΊln(𝐸𝑦𝑃+) , if 𝑦𝑃+ > 11.6 (9.1.49) In the above π‘˜π‘ƒβˆ— is evaluated at the previous iteration.

The rate of generation of turbulent kinetic energy G, under the Couette flow assumptions, is given by:

𝐺 ≃ πœπ‘€πœ•π‘‰π‘ƒ

πœ•π‘› (9.1.50)

Integration over the cell volume gives the π‘†π‘˜β€² part of the linearized source term (Gosman and Ideriah, 1976) as:

βˆ«π›Ώπ‘‰πΊ 𝑑𝑉 ≃ π‘†π‘˜β€² = πœπ‘€[𝑉𝑃 𝛿𝑛𝛿𝑉]

𝑃

(9.1.51) The transport equation for Ξ΅ is abandoned for the next-to-wall cells due to its inapplicability there. Instead, Ξ΅ is fixed to the value given by Eq. 9.1.47 by means of linearized source term, ie. by setting:

π‘†πœ€β€²= πΏπΆπœ‡3/4π‘˜π‘ƒ3/2 ΞΊ 𝑦𝑃 π‘†πœ€β€²β€²= 𝐿

(9.1.52) where L is a large number, typically 1030.

84

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