유체역학및열전달 chapter 4. basic equations of fluid flow...
TRANSCRIPT
부산대학교현규유체역학및열전달 1
유체역학및열전달
Chapter 4. Basic Equations of Fluid Flow (2)
-Momentum Balance Equation
부산대학교 화공생명공학부현 규 (Kyu Hyun)
부산대학교현규
Basic Equations of Fluid flow (1)
유체역학및열전달 2
Prediction of “viscoelastic” materials under deformation.
• General Balance equation :
• General Balance equation in case of flow :
Accumulation = Input- Output + generation - consumption
Rate of accumulation = rate of Input – rate of Output + rate of (generation – consumption)
Rate of accumulation = rate of Input – rate of Output + rate of (generation – consumption)
)()( timeareaquantityFlux
´=
부산대학교현규
Momentum Balance; Equation of motion (1)
(Rate of momentum Accumulation)= (Rate of momentum entering) – (Rate of momentum leaving)
+ (sum of Forces acting on the system)
(Rate of momentum Accumulation)= (Rate of momentum entering) – (Rate of momentum leaving)
+ (sum of Forces acting on the system)
유체역학및열전달 3
부산대학교현규
Momentum Balance; Equation of motion (1)
(Rate of momentum Accumulation)= (Rate of momentum entering) – (Rate of momentum leaving)
+ (sum of Forces acting on the system)
(Rate of momentum Accumulation)= (Rate of momentum entering) – (Rate of momentum leaving)
+ (sum of Forces acting on the system)
유체역학및열전달 4
Momentum은 벡터량이므로 (x,y,z), 3성분이 존재(surface) (direction)
Body force + Molecular force
부산대학교현규
Shear stress (1)
유체역학및열전달 5
yuAF m= A
F=t
-Shear stress
부산대학교현규
Ÿ Think back to the molecular picture from chemistryŸ The specifics of these forces, connections, and interactions must be
captured by the molecular forces term that we seek.
Shear stress (2)
유체역학및열전달 6
τMolecular stress?
(Face) (direction)
부산대학교현규
Momentum Balance; Equation of motion (2)
Momentum enters and leaves the volume element l partly by convection from flow of the bulk fluid l partly by viscous action as a result of the velocity gradients
Momentum enters and leaves the volume element l partly by convection from flow of the bulk fluid l partly by viscous action as a result of the velocity gradients
• Consider the flow rates of the x momentum (momentum flux) into and out of the volume element
• Convective flow of the bulk fluid:
• Momentum by molecular transport (viscous action) – Molecular force
• Force action on the system arise from pressure and the gravitational force
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zyxgppzy xxxx DDD+-DD D+ r)(
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유체역학및열전달 7
부산대학교현규
Momentum Balance; Equation of motion (3)
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(Eq. 4.20)
• 부피를 아주 작게 만들면
유체역학및열전달 8
부산대학교현규
운동방정식 (Equation of motion) (1)
• 운동방정식 (Eq. 4.20)을 Vector form으로 정리하면 다음과 같다.
• Convective flow momentum term을 정리하면 다음과 같다.• Momentum(3가지 방향)이 3가지 방향을 가지므로, 9가지 항이 존재
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유체역학및열전달 9
부산대학교현규
운동방정식 (Equation of motion) (2)• 운동방정식 (Eq. 4.20).
• Momentum by molecular transport (viscous action) 정리하면 다음과같다. [stress는 9가지가 존재한다 (3개의 면) x (3개의 방향)]
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유체역학및열전달 10
부산대학교현규
운동방정식 (Equation of motion) (3)• 따라서 위에서 구한 벡터와 텐서 항으로 부터 x,y,z component별로
정리하면 다음과 같다.
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유체역학및열전달 11
부산대학교현규
Equation of motion for Incompressible fluid (1)• 운동방정식 (Eq. 4.22): 앞의 방정식에서 연속방정식 이용, ρ가 일정
• 다른 component에도 적용가능 (y,z)
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-Continuity Equation
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Incompressible fluid
유체역학및열전달 12
부산대학교현규
Newton’s law of viscosity (1)
• Newtonian Constitutive equation (위의 실험적 발견을 확대 적용)• Stress tensor should be symmetric
유체역학및열전달 13
yu
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( )T)( VVτ Ñ+Ñ-= m
( )TVV Ñ+Ñ=g& -Rate-of strain tensor
부산대학교현규
Newton’s law of viscosity
유체역학및열전달 14
( )T)( VVτ Ñ+Ñ-= m
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부산대학교현규
Navier-Stokes Equations (1)• Equation of motion for a fluid of constant density and viscosity• 이 경우 stress와 velocity gradient와의 관계는 다음과 같다.
• 운동방정식에 대입하여 정리하면 다음과 같은 방정식을 얻을 수 있다.
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유체역학및열전달 15
부산대학교현규
Navier-Stokes Equations (2)• x 성분만 적용해보면 다음과 같다.
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유체역학및열전달 16
부산대학교현규
Navier-Stokes Equations (3)
• Navier Stokes 방정식을 벡터, 텐서 형태로 표현 하면
• Navier Stokes equation = Equation of motion for Newtonian Incompressible fluid
• Navier stokes equation in cylindrical coordinate and spherical coordinate는책을참조 eq. (4.33)~ (4.39)
• Euler equation• Equation of motion for a fluid of constant density and zero
viscosity (즉, potential flow 에 적용할수있는운동방정식)
gVV rmr +Ñ+-Ñ= 2pDtD
gV rr +-Ñ= pDtD
유체역학및열전달 17
부산대학교현규
Navier-Stokes Equations (4)
• Navier (Claude-Louis Navier) : French engineer and physicist
유체역학및열전달 18
• Stokes (Sir George Gabriel Stokes) : Mathematician and physicist
부산대학교현규
Equation of Motion
• Basic Equation of motion using Momentum Balance
• Mass Balance Equation (Continuity Eq’n) 적용 (Incompressible or compressible)
• Navier Stokes Equation (Newtonian Fluid) = μ (점도)가 일정
gτVVV rrr+Ñ-×Ñ-×-Ñ=
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gτV rr +×Ñ--Ñ= ][pDtD
gVV rmr +Ñ+-Ñ= 2pDtD
유체역학및열전달 19
( )T)( VVτ Ñ+Ñ-= m