Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic - PowerPoint PPT Presentation

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling T OMASZ G. Z IELI ´ NSKI (after: D.J. A CHESON ’s “ Elementary Fluid Dynamics ”) bluebox.ippt.pan.pl/˜tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw • Poland

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Outline Introduction 1 Mathematical preliminaries Basic notions and definitions Convective derivative

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Outline Introduction 1 Mathematical preliminaries Basic notions and definitions Convective derivative Ideal flow theory 2 Ideal fluid Incompressibility condition Euler’s equations of motion Boundary and interface-coupling conditions

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Outline Vorticity of flow 3 Introduction 1 Bernoulli theorems Mathematical preliminaries Vorticity Basic notions and Cylindrical flows definitions Rankine vortex Convective derivative Vorticity equation Ideal flow theory 2 Ideal fluid Incompressibility condition Euler’s equations of motion Boundary and interface-coupling conditions

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Outline Vorticity of flow 3 Introduction 1 Bernoulli theorems Mathematical preliminaries Vorticity Basic notions and Cylindrical flows definitions Rankine vortex Convective derivative Vorticity equation Ideal flow theory Basic aerodynamics 2 4 Ideal fluid Steady flow past a fixed Incompressibility condition wing Euler’s equations of Fluid circulation round a motion wing Boundary and Kutta–Joukowski theorem interface-coupling and condition conditions Concluding remarks

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Outline Vorticity of flow 3 Introduction 1 Bernoulli theorems Mathematical preliminaries Vorticity Basic notions and Cylindrical flows definitions Rankine vortex Convective derivative Vorticity equation Ideal flow theory Basic aerodynamics 2 4 Ideal fluid Steady flow past a fixed Incompressibility condition wing Euler’s equations of Fluid circulation round a motion wing Boundary and Kutta–Joukowski theorem interface-coupling and condition conditions Concluding remarks

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Mathematical preliminaries Theorem (Divergence theorem) Let the region V be bounded by a simple surface S with unit outward normal n . Then: � � � � f · n d S = ∇ · f d V ; in particular f n d S = ∇ f d V . S V S V

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Mathematical preliminaries Theorem (Divergence theorem) Let the region V be bounded by a simple surface S with unit outward normal n . Then: � � � � f · n d S = ∇ · f d V ; in particular f n d S = ∇ f d V . S V S V Theorem (Stokes’ theorem) Let C be a simple closed curve spanned by a surface S with unit normal n . Then: n � � � � f · d x = ∇ × f · n d S . S C C S Green’s theorem in the plane may be viewed as a special case of Stokes’ theorem � � (with f = u ( x , y ) , v ( x , y ) , 0 ): � ∂ v � � ∂ x − ∂ u � u d x + v d y = d x d y . ∂ y C S

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Basic notions and definitions A usual way of describing a fluid flow is by means of the flow velocity defined at any point x = ( x , y , z ) and at any time t : � � u = u ( x , t ) = u ( x , t ) , v ( x , t ) , w ( x , t ) . Here, u , v , w are the velocity components in Cartesian coordinates.

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Basic notions and definitions � � u = u ( x , t ) = u ( x , t ) , v ( x , t ) , w ( x , t ) Definition (Steady flow) A steady flow is one for which ∂ u � � ∂ t = 0 , that is, u = u ( x ) = u ( x ) , v ( x ) , w ( x ) .

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Basic notions and definitions � � u = u ( x , t ) = u ( x , t ) , v ( x , t ) , w ( x , t ) Definition (Steady flow) A steady flow is one for which ∂ u � � ∂ t = 0 , that is, u = u ( x ) = u ( x ) , v ( x ) , w ( x ) . Definition (Two-dimensional flow) A two-dimensional flow is of the form � � u = u ( x , t ) , v ( x , t ) , 0 where x = ( x , y ) .

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Basic notions and definitions � � u = u ( x , t ) = u ( x , t ) , v ( x , t ) , w ( x , t ) Definition (Steady flow) A steady flow is one for which ∂ u � � ∂ t = 0 , that is, u = u ( x ) = u ( x ) , v ( x ) , w ( x ) . Definition (Two-dimensional flow) A two-dimensional flow is of the form � � u = u ( x , t ) , v ( x , t ) , 0 where x = ( x , y ) . Definition (Two-dimensional steady flow) A two-dimensional steady flow is of the form � � u = u ( x ) , v ( x ) , 0 x = ( x , y ) . where

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Streamlines and particle paths (pathlines) Definition (Streamline) A streamline is a curve which, at any particular time t , has the same direction as u ( x , t ) at each point. A streamline x = x ( s ) , y = y ( s ) , z = z ( s ) ( s is a parameter) is obtained by solving at a particular time t : d y d x d z d s d s d s u = v = w .

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Streamlines and particle paths (pathlines) Definition (Streamline) A streamline is a curve which, at any particular time t , has the same direction as u ( x , t ) at each point. A streamline x = x ( s ) , y = y ( s ) , z = z ( s ) ( s is a parameter) is obtained by solving at a particular time t : d y d x d z d s d s d s u = v = w . For a steady flow the streamline pattern is the same at all times, and fluid particles travel along them. In an unsteady flow, streamlines and particle paths are usually quite different.

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Streamlines and particle paths (pathlines) Definition (Streamline) A streamline is a curve which, at any particular time t , has the same direction as u ( x , t ) at each point. A streamline x = x ( s ) , y = y ( s ) , z = z ( s ) ( s is a parameter) is obtained by solving at a particular time t : d y d x d z d s d s d s u = v = w . Example Consider a two-dimensional flow described as follows u ( x , t ) = u 0 , v ( x , t ) = a t , w ( x , t ) = 0 , where u 0 and a are positive constants. Now, notice that: in this flow streamlines are (always) straight lines, fluid particles follow parabolic paths: y ( t ) = a t 2 y = a x 2 x ( t ) = u 0 t , → 0 . 2 2 u 2

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Streamlines and particle paths (pathlines) y x t = 0 u ( x , t ) = u 0 , v ( x , t ) = a t , w ( x , t ) = 0 in this flow streamlines are (always) straight lines, fluid particles follow parabolic paths: y ( t ) = a t 2 y = a x 2 x ( t ) = u 0 t , → 0 . 2 2 u 2

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Streamlines and particle paths (pathlines) y t = ∆ t x u ( x , t ) = u 0 , v ( x , t ) = a t , w ( x , t ) = 0 in this flow streamlines are (always) straight lines, fluid particles follow parabolic paths: y ( t ) = a t 2 y = a x 2 x ( t ) = u 0 t , → 0 . 2 2 u 2

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Streamlines and particle paths (pathlines) y t = 2 ∆ t x u ( x , t ) = u 0 , v ( x , t ) = a t , w ( x , t ) = 0 in this flow streamlines are (always) straight lines, fluid particles follow parabolic paths: y ( t ) = a t 2 y = a x 2 x ( t ) = u 0 t , → 0 . 2 2 u 2

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Streamlines and particle paths (pathlines) y t = 2 ∆ t t = ∆ t x t = 0 u ( x , t ) = u 0 , v ( x , t ) = a t , w ( x , t ) = 0 in this flow streamlines are (always) straight lines, fluid particles follow parabolic paths: y ( t ) = a t 2 y = a x 2 x ( t ) = u 0 t , → 0 . 2 2 u 2

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Convective derivative Let f ( x , t ) denote some quantity of interest in the fluid motion.

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Convective derivative Let f ( x , t ) denote some quantity of interest in the fluid motion. Note that ∂ f ∂ t means the rate of change of f at a fixed position x in space.

Introduction Ideal flow theory Vorticity of flow Basic aerodynamics Convective derivative Let f ( x , t ) denote some quantity of interest in the fluid motion. Note that ∂ f ∂ t means the rate of change of f at a fixed position x in space. The rate of change of f “ following the fluid ” is D t = d d x D f = ∂ f d t + ∂ f � � x ( t ) , t d t f ∂ x ∂ t � � where x ( t ) = x ( t ) , y ( t ) , z ( t ) is understood to change with time at the local flow velocity u , so that � d x � d x d t , d y d t , d z � � d t = = u , v , w . d t Therefore, ∂ f d x d t = ∂ f d x d t + ∂ f d y d t + ∂ f d z d t ∂ x ∂ x ∂ y ∂ z