Introduction to Computational Fluid Dynamics Fluid Mechanics and Heat Transfer. Basic equations. Continuity (Mass Conservation) divv 0 div v 0 t incompressible Momentum Equation (Navier-Stokes equations) Incompressible viscous fluid ( constant, constant ) v v v F grad p v t Lecture 2 –Basic equations. Dimensionless parameters 1 Introduction to Computational Fluid Dynamics Energy equation dE d 1 p divk grad T dt dt where E is the internal energy of a fluid particle (viscous compressible fluid) - gas heating due to the compression d 1 p p divv dt continuity equation - mechanical work of the viscous forces transformation in heat Lecture 2 –Basic equations. Dimensionless parameters 2 Introduction to Computational Fluid Dynamics p By taking into account that the enthalpy of a unit of mass is i E we obtain di dp divk grad T dt dt But for a perfect gas we have di c pdT and energy equation becomes d dp c pT divk grad T dt dt where df Df f v f dt Dt t is the substantial derivative. Lecture 2 –Basic equations. Dimensionless parameters 3 Introduction to Computational Fluid Dynamics Momentum Equation (for fluid saturated porous media) By a porous medium we mean a material consisting of a solid matrix with an interconnected void. Porous media are present almost always in the surrounding medium, very few materials excepting fluids being non-porous. heat exchanger processor cooler porous rock Lecture 2 –Basic equations. Dimensionless parameters 4 Introduction to Computational Fluid Dynamics Darcy’s (law) equation (momentum equation) K v grad p g where v is the average velocity (filtration velocity, superficial velocity, seepage velocity or Darcian velocity) K k g is the permeability of the porous medium Darcy’ law expresses a linear dependence between the pressure gradient and the filtration velocity. It was reported that this linear law is not valid for large values of the pressure gradient. Lecture 2 –Basic equations. Dimensionless parameters 5 Introduction to Computational Fluid Dynamics Darcy’s law extensions Forchheimer’s extension c p v F v v K K where cF is a dimensionless parameter depending on the porous medium Brinkman’s extension p v ~ v K where ~ is an effective viscosity Brinkman-Forchheimer’s extension c p v F v v ~ v K K Lecture 2 –Basic equations. Dimensionless parameters 6 Introduction to Computational Fluid Dynamics Energy equation for porous media T c c v T k T p m p fluid e t where c c 1 c p m p fluid p solid ke k fluid 1ksolid Lecture 2 –Basic equations. Dimensionless parameters 7 Introduction to Computational Fluid Dynamics Bouyancy driven flow1 There exists a large class of fluid flows in which the motion is caused by buoyancy in the fluid. Buoyancy is the force experienced in a fluid due to a variation of density in the presence of a gravitational field. According to the definition of an incompressible fluid, variations in the density normally mean that the fluid is compressible, rather than incompressible. For many of the fluid flows of the type mentioned above, the density variation is important only in the body-force term of the momentum equations. In all other places in which the density appears in the governing equations, the variation of density leads to an insignificant effect. Buoyancy results in a force acting on the fluid, and the fluid would accelerate continuously if it were not for the existence of the viscous forces. The situation depicted above occurs in natural convection. 1 I.G. Currie, Fundamental Mechanics of Fluids, 3rd ed., Marcel Dekker, New York, 2003 Lecture 2 –Basic equations. Dimensionless parameters 8 Introduction to Computational Fluid Dynamics Generally, density is a function of temperature and concentration T,c Black Sea salinity, temperature and density (Ecological Modelling, 221, 2010, p. 2287-2301) Lecture 2 –Basic equations. Dimensionless parameters 9 Introduction to Computational Fluid Dynamics Boussinesq’s approximation , Momentum Lecture 2 –Basic equations. Dimensionless parameters 10 Introduction to Computational Fluid Dynamics Dimensionless equations u v w 0 x y z u u u u p 2u 2u 2u u v w F x 2 2 2 t x y z x x y z v v v v p 2v 2v 2v u v w F y 2 2 2 t x y z y x y z w w w w p 2w 2w 2w u v w F z 2 2 2 t x y z z x y z T T T T 2T 2T 2T c u v w k q( 0) p 2 2 2 t x y z x y z 2 2 2 2 2 2 u v w v u w v u w 2 2 2 x x x x y y z z x Lecture 2 –Basic equations. Dimensionless parameters 11 Introduction to Computational Fluid Dynamics We introduce further the dimensionless variables t x y z u v w F p T T , X , Y , Z , U , V , W , F ' , P , 0 t0 L L L U0 U0 U0 g p0 T into the governing equations u U0 U u X U0 U UU0 , UU0 , t t t0 x X x L X p X p P P p 0 , x X 0 x L X 2u u U U U 2U X U 2U 0 0 0 x2 x x x L X L X 2 x L2 X 2 T T T X T T T0 , T T0 t t t0 x X x L X 2T T T T 2 X T 2U x2 x x x L X L X 2 x L2 X 2 Lecture 2 –Basic equations. Dimensionless parameters 12 Introduction to Computational Fluid Dynamics and we obtain the dimensionless equations U V W 0 X Y Z L U U V W gL p P 2U 2U 2U U V W F ' 0 2 x 2 2 2 2 t0U0 X Y Z U0 U0 X U0L X Y Z k 2 2 2 L Q c p U U V W 0 2 2 2 t0U0 X Y Z U0L X Z Z c p LT Lecture 2 –Basic equations. Dimensionless parameters 13 Introduction to Computational Fluid Dynamics Dimensionless numbers A. Strouhal number The Strouhal number (St) is a dimensionless number describing oscillating flow mechanisms: L L oscillation St t0U0 U0 average velocity B. Froude number The Froude number (Fr) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). In naval architecture the Froude number is a very significant figure used to determine the resistance of a partially submerged object moving through water, and permits the comparison of similar objects of different sizes, because the wave pattern generated is similar at the same Froude number only. U Fr 0 gL Lecture 2 –Basic equations. Dimensionless parameters 14 Introduction to Computational Fluid Dynamics C. Euler number It expresses the relationship between a local pressure drop e.g. over a restriction and the kinetic energy per volume, and is used to characterize losses in the flow, where a perfect frictionless flow corresponds to an Euler number of 1. p0 Eu 2 U0 2 Usually, we take p0 U0 and we obtain Eu = 1. D. Reynolds number U L U L Re 0 0 The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion; turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities. Lecture 2 –Basic equations. Dimensionless parameters 15 Introduction to Computational Fluid Dynamics E. Prandtl number c / Pr p k k / c p It is the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number contains no such length scale in its definition and is dependent only on the fluid and the fluid state. As such, the Prandtl number is often found in property tables alongside other properties such as viscosity and thermal conductivity. •gases - Pr ranges 0.7 - 1.0 •water - Pr ranges 1 - 10 •liquid metals - Pr ranges 0.001 - 0.03 •oils - Pr ranges 50 - 2000 E. Eckert number It expresses the relationship between a flow's kinetic energy and the boundary layer enthalpy difference, and is used to characterize heat dissipation. U 2 Εc 0 c p T Lecture 2 –Basic equations. Dimensionless parameters 16 Introduction to Computational Fluid Dynamics Using the above dimensionless numbers we obtain: U U V W 1 P 1 2U 2U 2U St U V W F 'x Eu X Y ZU FrV W X Re X 2 Y 2 Z 2 0 X Y…. Z Q 1 2 2 2 Ec St U V W 2 2 2 X Y Z Re Pr X Z Z Re or using differential operators V 0 (23) V 1 1 St V V F'Eu P V (24) Fr Re 1 Ec St V (25) Re Pr Re Lecture 2 –Basic equations. Dimensionless parameters 17 Introduction to Computational Fluid Dynamics Comments Dimensionless numbers allow for comparisons between very different systems and tell you how the system will behave Many useful relationships exist between dimensionless numbers that tell you how specific things influence the system When you need to solve a problem numerically, dimensionless groups help you to scale your problem. Analytical studies can be performed for limiting values of the dimensionless parameters Express the governing equations in dimensionless form to: (1) identify the governing parameters (2) plan experiments (3) guide in the presentation of experimental results and theoretical solutions If the flow is not oscillatory, we take usually t0 L U0 and thus St =1.