# Fluid Mechanics Fluid Kinematics Velocity Field Flow

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FLUID MECHANICS

FLUID KINEMATICS

Fluid Kinematics gives the geometry of fluid motion. It is a branch of fluid mechanics, which describes the fluid motion, and it’s consequences without consideration of the nature of forces causing the motion. Fluid kinematics is the study of velocity as a function of space and time in the flow field. From velocity, pressure variations and hence, forces acting on the fluid can be determined.

VELOCITY FIELD

Velocity at a given point is defined as the instantaneous velocity of the fluid particle, which at a given instant is passing through the point. It is represented by V=V(x,y,z,t). Vectorially, V=ui+vj+wk where u,v,w are three scalar components of velocity in x,y and z directions and (t) is the time. Velocity is a vector quantity and velocity field is a vector field.

FLOW PATTERNS

Fig. Flow Patterns Fluid Mechanics is a visual subject. Patterns of flow can be visualized in several ways. Basic types of line patterns used to visualize flow are streamline, path line, streak line and time line. (a) Stream line is a line, which is everywhere tangent to the velocity vector at a given instant. (b) Path line is the actual path traversed by a given particle. (c) Streak line is the locus of particles that have earlier passed through a prescribed point. (d) Time line is a set of fluid particles that form a line at a given instant.

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Streamline is convenient to calculate mathematically. Other three lines are www.getmyuneais.iecrotmo obtain experimentally. Streamlines are difficult to generate experimentally.

Streamlines and Time lines are instantaneous lines. Path lines and streak lines are generated by passage of time. In a steady flow situation, streamlines, path lines and streak lines are identical. In Fluid Mechanics, the most common mathematical result for flow visualization is the streamline pattern – It is a common method of flow pattern presentation.

Streamlines are everywhere tangent to the local velocity vector. For a stream line, (dx/u) = (dy/v) = (dz/w). Stream tube is formed by a closed collection of streamlines. Fluid within the stream tube is confined there because flow cannot cross streamlines. Stream tube walls need not be solid, but may be fluid surfaces

METHOD OF DESCRIBING FLUID MOTION

Two methods of describing the fluid motion are: (a) Lagrangian method and (b) Eularian method.

Fig. Lagrangian method A single fluid particle is followed during its motion and its velocity, acceleration etc. are described with respect to time. Fluid motion is described by tracing the kinematics behavior of each and every individual particle constituting the flow. We follow individual fluid particle as it moves through the flow. The particle is identified by its position at some instant and the time elapsed since that instant. We identify and follow small, fixed masses of fluid. To describe the fluid flow where there is a relative motion, we need to follow many particles and to resolve details of the flow; we need a large number of particles. Therefore, Langrangian method is very difficult and not widely used in Fluid Mechanics.

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EULARIAN METHOD

Fig. Eulerian Method

The velocity, acceleration, pressure etc. are described at a point or at a section as a function of time. This method commonly used in Fluid Mechanics. We look for field description, for Ex.; seek the velocity and its variation with time at each and every location in a flow field. Ex., V=V(x,y,z,t). This is also called control volume approach. We draw an imaginary box around a fluid system. The box can be large or small, and it can be stationary or in motion.

TYPES OF FLUID FLOW

1. Steady and Un-steady flows 2. Uniform and Non-uniform flows 3. Laminar and Turbulent flows 4. Compressible and Incompressible flows 5. Rotational and Irrotational flows 6. One, Two and Three dimensional flows

STEADY AND UNSTEADY FLOW

Steady flow is the type of flow in which the various flow parameters and fluid properties at any point do not change with time. In a steady flow, any property may vary from point to point in the field, but all properties remain constant with time at every point.[∂V/∂ t] x,y,z= 0; [∂p/ ∂t]x,y,z =0. Ex.: V=V(x,y,z); p=p(x,y,z) . Time is a criterion.

Unsteady flow is the type of flow in which the various flow parameters and fluid properties at any point change with time. [∂V/∂t]x,y,z≠0 ; [∂p/∂t]x,y,z≠0, Eg.:V=V(x,y,z,t), p=p(x,y,z,t) or V=V(t), p=p(t) . Time is a criterion

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UNIFORM AND NON-UNIFORM FLOWS

www.getmyuni.com Uniform Flow is the type of flow in which velocity and other flow parameters at

any instant of time do not change with respect to space. Eg., V=V(x) indicates that the flow is uniform in ‘y’ and ‘z’ axis. V=V (t) indicates that the flow is uniform in ‘x’, ‘y’ and ‘z’ directions. Space is a criterion.

Uniform flow field is used to describe a flow in which the magnitude and direction of the velocity vector are constant, i.e., independent of all space coordinates throughout the entire flow field (as opposed to uniform flow at a cross section). That is, [∂V/ ∂s]t=constant =0, that is ‘V’ has unique value in entire flow field

Non-uniform flow is the type of flow in which velocity and other flow parameters at any instant change with respect to space. [∂V/ ∂s]t=constant is not equal to zero. Distance or space is a criterion

LAMINAR AND TURBULANT FLOWS

Laminar Flow is a type of flow in which the fluid particles move along welldefined paths or stream-lines. The fluid particles move in laminas or layers gliding smoothly over one another. The behavior of fluid particles in motion is a criterion.

Turbulent Flow is a type of flow in which the fluid particles move in zigzag way in the flow field. Fluid particles move randomly from one layer to another. Reynolds number is a criterion. We can assume that for a flow in pipe, for Reynolds No. less than 2000, the flow is laminar; between 2000-4000, the flow is transitional; and greater than 4000, the flow is turbulent.

COMPRESSIBLE AND INCOMPRESSIBLE FLOWS

Incompressible Flow is a type of flow in which the density (ρ) is constant in the flow field. This assumption is valid for flow Mach numbers with in 0.25. Mach number is used as a criterion. Mach Number is the ratio of flow velocity to velocity of sound waves in the fluid medium

Compressible Flow is the type of flow in which the density of the fluid changes in the flow field. Density is not constant in the flow field. Classification of flow based on Mach number is given below:

M < 0.25 – Low speed M < unity – Subsonic M around unity – Transonic M > unity – Supersonic M > > unity, (say 7) – Hypersonic

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www.getmyuni.com ROTATIONAL AND IRROTATIONAL FLOWS

Rotational flow is the type of flow in which the fluid particles while flowing along stream-lines also rotate about their own axis.

Ir-rotational flow is the type of flow in which the fluid particles while flowing along stream-lines do not rotate about their own axis.

ONE,TWO AND THREE DIMENSIONAL FLOWS

The number of space dimensions needed to define the flow field completely governs dimensionality of flow field. Flow is classified as one, two and threedimensional depending upon the number of space co-ordinates required to specify the velocity fields.

One-dimensional flow is the type of flow in which flow parameters such as velocity is a function of time and one space coordinate only. For Ex., V=V(x,t) – 1-D, unsteady ; V=V(x) – 1-D, steady

Two-dimensional flow is the type of flow in which flow parameters describing the flow vary in two space coordinates and time. For Ex., V=V(x,y,t) – 2-D, unsteady; V=V(x,y) – 2-D, steady

Three-dimensional flow is the type of flow in which the flow parameters describing the flow vary in three space coordinates and time. For Ex., V=V(x,y,z,t) – 3-D, unsteady ; V=V(x,y,z) – 3D, steady

CONTINUITY EQUATION

Rate of flow or discharge (Q) is the volume of fluid flowing per second. For incompressible fluids flowing across a section, Volume flow rate, Q= A×V m3/s where A=cross sectional area and V= average velocity. For compressible fluids, rate of flow is expressed as mass of fluid flowing across a section per second. Mass flow rate (m) =(ρAV) kg/s where ρ = density.

Fig. Continuity Equation

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www.getmyuni.com Continuity equation is based on Law of Conservation of Mass. For a fluid flowing through a pipe, in a steady flow, the quantity of fluid flowing per second at all crosssections is a constant. Let v1=average velocity at section [1], ρ1=density of fluid at [1], A1=area of flow at [1]; Let v2, ρ2, A2 be corresponding values at section [2]. Rate of flow at section [1]= ρ1 A1 v1 Rate of flow at section [2]= ρ2 A2 v2 ρ1 A1 v1= ρ2 A2 v2 This equation is applicable to steady compressible or incompressible fluid flows and is called Continuity Equation. If the fluid is incompressible, ρ1 = ρ2 and the continuity equation reduces to A1 v1= A2 v2 For steady, one dimensional flow with one inlet and one outlet, ρ 1 A1 v1 − ρ2 A2 v2=0 For control volume with N inlets and outlets ∑i=1N (ρi Ai vi) =0 where inflows are positive and outflows are negative . Velocities are normal to the areas. This is the continuity equation for steady one dimensional flow through a fixed control volume When density is constant, ∑i=1N (Ai vi) =0

Problem 1 The diameters of the pipe at sections (1) and (2) are 15cm and 20cm respectively. Find the discharge through the pipe if the velocity of water at section (1) is 4m/s. Determine also the velocity at section (2)

(Answers: 0.0706m3/s, 2.25m/s)

Problem-2 A 40cm diameter pipe conveying water branches into two pipes of diameters 30cm and 20cm respectively. If the average velocity in the 40cm diameter pipe is 3m/s., find the discharge in this pipe. Also, determine the velocity in 20cm diameter pipe if the average velocity in 30cm diameter pipe is 2m/s. (Answers: 0.3769m3/s., 7.5m/s.)

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CONTINUITY EQUATION IN 3-DIMENSIONS (Differential form, Cartesian co-ordinates)

Consider infinitesimal control volume as shown of dimensions dx, dy and dz in x,y,and z directions

Fig. Continuity Equation in Three Dimensions

u,v,w are the velocities in x,y,z directions. Mass of fluid entering the face ABCD = Density ×velocity in x-direction ×Area ABCD = ρudy.dz Mass of fluid leaving the face EFGH= ρudy.dz+ [∂ (ρudy.dz)/ ∂x](dx) Therefore, net rate of mass efflux in x-direction= −[∂ (ρudy.dz)/ ∂x](dx) = - [∂ (ρu)/ ∂ x](dxdydz) Similarly, the net rate of mass efflux in y-direction= - [∂ (ρv)/ ∂y](dxdydz) z-direction= - [∂ (ρw)/ ∂z](dxdydz) The rate of accumulation of mass within the control volume =∂ (ρdV)/ ∂t = ρ∂/∂t (dV) where dV=Volume of the element=dxdydz and dV is invarient with time. From conservation of mass, the net rate of efflux = Rate of accumulation of mass within the control volume. - [∂(ρu)/ ∂ x + ∂ (ρv)/ ∂ y + ∂ (ρw)/ ∂ z ](dxdydz) = ρ∂/∂t(dxdydz) OR ρ∂/∂t + ∂ (ρu)/ ∂ x +∂ (ρv)/ ∂ y + ∂ (ρw)/ ∂ z =0

This is the continuity equation applicable for (a) Steady and unsteady flows (b) Uniform and non-uniform flows (c) Compressible and incompressible flows. For steady flows, (∂/∂t) = 0 and [∂ (ρu) /∂x + ∂ (ρv)/ ∂y + ∂ (ρw)/ ∂z] =0 If the fluid is incompressible, ρ= constant [∂u/∂x + ∂v/∂y + ∂w/∂z] =0 This is the continuity equation for 3-D flows. For 2-D flows, w=0 and [∂u/∂x + ∂v/∂y] = 0

VELOCITY AND ACCELERATION

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Let V= Resultant velocity at any point in a fluid flow. (u,v,w) are the velocity components in x,y and z directions which are functions of space coordinates and time. www.getmyuun=iu.(cx,oym,z,t) ; v=v(x,y,z,t) ;w=w(x,y,z,t). Resultant velocity=V=ui+vj+wk |V|= (u2+v2+w2)1/2 Let ax,ay,az are the total accelerations in the x,y,z directions respectively ax= [du/ dt]= [∂u/ ∂x] [∂x/ ∂t]+ [∂u/ ∂y][ ∂y/ ∂t]+ [∂u/ ∂z][ ∂z/ ∂t]+ [∂u/ ∂t] ax= [du/dt] =u[∂u/ ∂x]+v[∂u/ ∂y]+w[∂u/ ∂z]+[ ∂u/∂ t]

Similarly, ay= [dv/ dt] = u[∂v/ ∂x]+ v[∂v/ ∂y]+w[∂v/ ∂z]+[ ∂v/∂t] az= [dw/dt] = u[∂w/ ∂x]+ v[∂w/ ∂y]+w[∂w/ ∂z]+[ ∂w/ ∂t]

1. Convective Acceleration Terms – The first three terms in the expressions for ax, ay, az. Convective acceleration is defined as the rate of change of velocity due to change of position of the fluid particles in a flow field

2. Local Acceleration Terms- The 4th term, [∂ ( )/ ∂t] in the expressions for ax, ay, az. Local or temporal acceleration is the rate of change of velocity with respect to time at a given point in a flow field.

Material or Substantial Acceleration = Convective Acceleration + Local or Temporal Acceleration. In a steady flow, temporal or local acceleration is zero. In uniform flow, convective acceleration is zero.

For steady flow, [∂u/ ∂t]= [∂v/ ∂t]= [∂w/ ∂t]= 0 ax= [du/dt]= u[∂u/∂x]+v[∂u/∂y]+w[∂u/∂z] ay= [dv/dt]= u[∂v/∂x]+v[∂v/∂y]+w[∂v/∂z] az= [dw/dt]= u[∂w/∂x]+v[∂w/∂y]+w[∂w/∂z] Acceleration Vector=axi+ayj+azk; |A|=[ax2+aY2+az2]1/2

Problem 1 The fluid flow field is given by V=x2yi+y2zj-(2xyz+yz2)k . Prove that this is a case of a possible steady incompressible flow field. u=x2y; v=y2z; w= -2xyz -yz2 (∂u/∂x) = 2xy; (∂v/∂y)=2yz; (∂w/∂z)= -2xy -2yz For steady incompressible flow, the continuity equation is [∂u/∂x + ∂v/∂y + ∂w/∂z] =0 2xy + 2yz -2xy -2yz =0 Therefore, the given flow field is a possible case of steady incompressible fluid flow.

Problem-2.

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Given v=2y2 and w=2xyz, the two velocity components. Determine the third component such that it satisfies the continuity equation. v=2y2; w=2xyz; (∂v/∂y)=4y; (∂w/∂z)=2xy www.getmyu(n∂iu./∂cxo) m+ (∂v/∂y) + (∂w/∂z) =0 (∂u/∂x) = -4y -2xy; ∂u = (-4y -2xy) ∂x u= -4xy -x2y+ f(y,z); f(y,z) can not be the function of (x)

Problem-3. Find the acceleration components at a point (1,1,1) for the following flow field: u=2x2+3y; v= -2xy+3y2+3zy; w= -(3/2)z2+2xz -9y2z ax= [∂u/∂t]+ u[∂u/∂x]+v[∂u/∂y]+w[∂u/∂z] 0+(2x2+3y)4x+(-2xy+3y2+3zy)3+0 ; [ax] 1,1,1= 32 units Similarly, ay=[∂v/∂t]+ u[∂v/∂x]+v[∂v/∂y]+w[∂v/∂z] ay = 0+(2x2+3y) (-2x)+ (-2xy+3y2+3zy)(-2x+6y+3z)+ {-(3/2)z2+2xz -9y2z}3y [ay]1,1,1 = -7.5 units Similarly, az= [∂w/∂t]+ u[∂w/∂x]+v[∂w/∂y]+w[∂w/ ∂z] Substituting, az= 23 units Resultant |a|= (ax2+ay2+az2)1/2 Problem-4. Given the velocity field V= (4+xy+2t)i + 6x3j + (3xt2+z)k. Find acceleration of a fluid particle at (2,4,-4) at t=3. [dV/dt]=[ ∂V/∂t]+u[∂V/∂x]+v[∂V/∂y]+w[∂V/∂z] u= (4+xy+2t); v=6x3; w= (3xt2+z) [∂V/∂x]= (yi+18x2j+3t2k); [∂V/∂y]= xi; [∂V/∂z]= k; [∂V/∂t] = 2i+6xtk . Substituting, [dV/dt]= (2+4y+xy2+2ty+6x4)i + (72x2+18x3y+36tx2)j + (6xt+12t2+3xyt2+6t3+z+3xt2)k The acceleration vector at the point (2,4,-4) and time t=3 is obtained by substitution, a= 170i+1296j+572k; Therefore, ax=170, ay=1296, az=572 Resultant |a|= [1702+12962+5722]1/2 units = 1426.8 units.

VELOCITY POTENTIAL AND STREAM FUNCTION

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Velocity Potential Function is a Scalar Function of space and time co-ordinates such that its negative derivatives with respect to any direction give the fluid velocity in that direction. www.getmyuΦni=.cΦo(mx,y,z) for steady flow. u= -(∂Φ/∂x); v= -(∂Φ/∂y); w= -(∂Φ/∂z) where u,v,w are the components of velocity in x,y and z directions. In cylindrical co-ordinates, the velocity potential function is given by ur= (∂Φ/∂r), uθ = (1/r)( ∂Φ/∂θ) The continuity equation for an incompressible flow in steady state is (∂u/∂x + ∂v/∂y + ∂w/∂z) = 0 Substituting for u, v and w and simplifying, (∂2Φ /∂x2 + ∂2Φ /∂y2 + ∂2Φ/∂z2) = 0 Which is a Laplace Equation. For 2-D Flow, (∂2Φ /∂x2 + ∂2Φ /∂y2) =0 If any function satisfies Laplace equation, it corresponds to some case of steady incompressible fluid flow.

IRROTATIONAL FLOW AND VELOCITY POTENTIAL

Assumption of Ir-rotational flow leads to the existence of velocity potential. Consider the rotation of the fluid particle about an axis parallel to z-axis. The rotation component is defined as the average angular velocity of two infinitesimal linear segments that are mutually perpendicular to each other and to the axis of rotation. Consider two-line segments δx, δy. The particle at P(x,y) has velocity components u,v in the x-y plane.

Fig. Rotation of a fluid partical.

The angular velocities of δx and δy are sought. The angular velocity of (δx) is {[v+ (∂v/∂x) δx –v] / δx} = (∂v/∂x) rad/sec

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FLUID MECHANICS

FLUID KINEMATICS

Fluid Kinematics gives the geometry of fluid motion. It is a branch of fluid mechanics, which describes the fluid motion, and it’s consequences without consideration of the nature of forces causing the motion. Fluid kinematics is the study of velocity as a function of space and time in the flow field. From velocity, pressure variations and hence, forces acting on the fluid can be determined.

VELOCITY FIELD

Velocity at a given point is defined as the instantaneous velocity of the fluid particle, which at a given instant is passing through the point. It is represented by V=V(x,y,z,t). Vectorially, V=ui+vj+wk where u,v,w are three scalar components of velocity in x,y and z directions and (t) is the time. Velocity is a vector quantity and velocity field is a vector field.

FLOW PATTERNS

Fig. Flow Patterns Fluid Mechanics is a visual subject. Patterns of flow can be visualized in several ways. Basic types of line patterns used to visualize flow are streamline, path line, streak line and time line. (a) Stream line is a line, which is everywhere tangent to the velocity vector at a given instant. (b) Path line is the actual path traversed by a given particle. (c) Streak line is the locus of particles that have earlier passed through a prescribed point. (d) Time line is a set of fluid particles that form a line at a given instant.

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Streamline is convenient to calculate mathematically. Other three lines are www.getmyuneais.iecrotmo obtain experimentally. Streamlines are difficult to generate experimentally.

Streamlines and Time lines are instantaneous lines. Path lines and streak lines are generated by passage of time. In a steady flow situation, streamlines, path lines and streak lines are identical. In Fluid Mechanics, the most common mathematical result for flow visualization is the streamline pattern – It is a common method of flow pattern presentation.

Streamlines are everywhere tangent to the local velocity vector. For a stream line, (dx/u) = (dy/v) = (dz/w). Stream tube is formed by a closed collection of streamlines. Fluid within the stream tube is confined there because flow cannot cross streamlines. Stream tube walls need not be solid, but may be fluid surfaces

METHOD OF DESCRIBING FLUID MOTION

Two methods of describing the fluid motion are: (a) Lagrangian method and (b) Eularian method.

Fig. Lagrangian method A single fluid particle is followed during its motion and its velocity, acceleration etc. are described with respect to time. Fluid motion is described by tracing the kinematics behavior of each and every individual particle constituting the flow. We follow individual fluid particle as it moves through the flow. The particle is identified by its position at some instant and the time elapsed since that instant. We identify and follow small, fixed masses of fluid. To describe the fluid flow where there is a relative motion, we need to follow many particles and to resolve details of the flow; we need a large number of particles. Therefore, Langrangian method is very difficult and not widely used in Fluid Mechanics.

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EULARIAN METHOD

Fig. Eulerian Method

The velocity, acceleration, pressure etc. are described at a point or at a section as a function of time. This method commonly used in Fluid Mechanics. We look for field description, for Ex.; seek the velocity and its variation with time at each and every location in a flow field. Ex., V=V(x,y,z,t). This is also called control volume approach. We draw an imaginary box around a fluid system. The box can be large or small, and it can be stationary or in motion.

TYPES OF FLUID FLOW

1. Steady and Un-steady flows 2. Uniform and Non-uniform flows 3. Laminar and Turbulent flows 4. Compressible and Incompressible flows 5. Rotational and Irrotational flows 6. One, Two and Three dimensional flows

STEADY AND UNSTEADY FLOW

Steady flow is the type of flow in which the various flow parameters and fluid properties at any point do not change with time. In a steady flow, any property may vary from point to point in the field, but all properties remain constant with time at every point.[∂V/∂ t] x,y,z= 0; [∂p/ ∂t]x,y,z =0. Ex.: V=V(x,y,z); p=p(x,y,z) . Time is a criterion.

Unsteady flow is the type of flow in which the various flow parameters and fluid properties at any point change with time. [∂V/∂t]x,y,z≠0 ; [∂p/∂t]x,y,z≠0, Eg.:V=V(x,y,z,t), p=p(x,y,z,t) or V=V(t), p=p(t) . Time is a criterion

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UNIFORM AND NON-UNIFORM FLOWS

www.getmyuni.com Uniform Flow is the type of flow in which velocity and other flow parameters at

any instant of time do not change with respect to space. Eg., V=V(x) indicates that the flow is uniform in ‘y’ and ‘z’ axis. V=V (t) indicates that the flow is uniform in ‘x’, ‘y’ and ‘z’ directions. Space is a criterion.

Uniform flow field is used to describe a flow in which the magnitude and direction of the velocity vector are constant, i.e., independent of all space coordinates throughout the entire flow field (as opposed to uniform flow at a cross section). That is, [∂V/ ∂s]t=constant =0, that is ‘V’ has unique value in entire flow field

Non-uniform flow is the type of flow in which velocity and other flow parameters at any instant change with respect to space. [∂V/ ∂s]t=constant is not equal to zero. Distance or space is a criterion

LAMINAR AND TURBULANT FLOWS

Laminar Flow is a type of flow in which the fluid particles move along welldefined paths or stream-lines. The fluid particles move in laminas or layers gliding smoothly over one another. The behavior of fluid particles in motion is a criterion.

Turbulent Flow is a type of flow in which the fluid particles move in zigzag way in the flow field. Fluid particles move randomly from one layer to another. Reynolds number is a criterion. We can assume that for a flow in pipe, for Reynolds No. less than 2000, the flow is laminar; between 2000-4000, the flow is transitional; and greater than 4000, the flow is turbulent.

COMPRESSIBLE AND INCOMPRESSIBLE FLOWS

Incompressible Flow is a type of flow in which the density (ρ) is constant in the flow field. This assumption is valid for flow Mach numbers with in 0.25. Mach number is used as a criterion. Mach Number is the ratio of flow velocity to velocity of sound waves in the fluid medium

Compressible Flow is the type of flow in which the density of the fluid changes in the flow field. Density is not constant in the flow field. Classification of flow based on Mach number is given below:

M < 0.25 – Low speed M < unity – Subsonic M around unity – Transonic M > unity – Supersonic M > > unity, (say 7) – Hypersonic

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www.getmyuni.com ROTATIONAL AND IRROTATIONAL FLOWS

Rotational flow is the type of flow in which the fluid particles while flowing along stream-lines also rotate about their own axis.

Ir-rotational flow is the type of flow in which the fluid particles while flowing along stream-lines do not rotate about their own axis.

ONE,TWO AND THREE DIMENSIONAL FLOWS

The number of space dimensions needed to define the flow field completely governs dimensionality of flow field. Flow is classified as one, two and threedimensional depending upon the number of space co-ordinates required to specify the velocity fields.

One-dimensional flow is the type of flow in which flow parameters such as velocity is a function of time and one space coordinate only. For Ex., V=V(x,t) – 1-D, unsteady ; V=V(x) – 1-D, steady

Two-dimensional flow is the type of flow in which flow parameters describing the flow vary in two space coordinates and time. For Ex., V=V(x,y,t) – 2-D, unsteady; V=V(x,y) – 2-D, steady

Three-dimensional flow is the type of flow in which the flow parameters describing the flow vary in three space coordinates and time. For Ex., V=V(x,y,z,t) – 3-D, unsteady ; V=V(x,y,z) – 3D, steady

CONTINUITY EQUATION

Rate of flow or discharge (Q) is the volume of fluid flowing per second. For incompressible fluids flowing across a section, Volume flow rate, Q= A×V m3/s where A=cross sectional area and V= average velocity. For compressible fluids, rate of flow is expressed as mass of fluid flowing across a section per second. Mass flow rate (m) =(ρAV) kg/s where ρ = density.

Fig. Continuity Equation

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www.getmyuni.com Continuity equation is based on Law of Conservation of Mass. For a fluid flowing through a pipe, in a steady flow, the quantity of fluid flowing per second at all crosssections is a constant. Let v1=average velocity at section [1], ρ1=density of fluid at [1], A1=area of flow at [1]; Let v2, ρ2, A2 be corresponding values at section [2]. Rate of flow at section [1]= ρ1 A1 v1 Rate of flow at section [2]= ρ2 A2 v2 ρ1 A1 v1= ρ2 A2 v2 This equation is applicable to steady compressible or incompressible fluid flows and is called Continuity Equation. If the fluid is incompressible, ρ1 = ρ2 and the continuity equation reduces to A1 v1= A2 v2 For steady, one dimensional flow with one inlet and one outlet, ρ 1 A1 v1 − ρ2 A2 v2=0 For control volume with N inlets and outlets ∑i=1N (ρi Ai vi) =0 where inflows are positive and outflows are negative . Velocities are normal to the areas. This is the continuity equation for steady one dimensional flow through a fixed control volume When density is constant, ∑i=1N (Ai vi) =0

Problem 1 The diameters of the pipe at sections (1) and (2) are 15cm and 20cm respectively. Find the discharge through the pipe if the velocity of water at section (1) is 4m/s. Determine also the velocity at section (2)

(Answers: 0.0706m3/s, 2.25m/s)

Problem-2 A 40cm diameter pipe conveying water branches into two pipes of diameters 30cm and 20cm respectively. If the average velocity in the 40cm diameter pipe is 3m/s., find the discharge in this pipe. Also, determine the velocity in 20cm diameter pipe if the average velocity in 30cm diameter pipe is 2m/s. (Answers: 0.3769m3/s., 7.5m/s.)

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CONTINUITY EQUATION IN 3-DIMENSIONS (Differential form, Cartesian co-ordinates)

Consider infinitesimal control volume as shown of dimensions dx, dy and dz in x,y,and z directions

Fig. Continuity Equation in Three Dimensions

u,v,w are the velocities in x,y,z directions. Mass of fluid entering the face ABCD = Density ×velocity in x-direction ×Area ABCD = ρudy.dz Mass of fluid leaving the face EFGH= ρudy.dz+ [∂ (ρudy.dz)/ ∂x](dx) Therefore, net rate of mass efflux in x-direction= −[∂ (ρudy.dz)/ ∂x](dx) = - [∂ (ρu)/ ∂ x](dxdydz) Similarly, the net rate of mass efflux in y-direction= - [∂ (ρv)/ ∂y](dxdydz) z-direction= - [∂ (ρw)/ ∂z](dxdydz) The rate of accumulation of mass within the control volume =∂ (ρdV)/ ∂t = ρ∂/∂t (dV) where dV=Volume of the element=dxdydz and dV is invarient with time. From conservation of mass, the net rate of efflux = Rate of accumulation of mass within the control volume. - [∂(ρu)/ ∂ x + ∂ (ρv)/ ∂ y + ∂ (ρw)/ ∂ z ](dxdydz) = ρ∂/∂t(dxdydz) OR ρ∂/∂t + ∂ (ρu)/ ∂ x +∂ (ρv)/ ∂ y + ∂ (ρw)/ ∂ z =0

This is the continuity equation applicable for (a) Steady and unsteady flows (b) Uniform and non-uniform flows (c) Compressible and incompressible flows. For steady flows, (∂/∂t) = 0 and [∂ (ρu) /∂x + ∂ (ρv)/ ∂y + ∂ (ρw)/ ∂z] =0 If the fluid is incompressible, ρ= constant [∂u/∂x + ∂v/∂y + ∂w/∂z] =0 This is the continuity equation for 3-D flows. For 2-D flows, w=0 and [∂u/∂x + ∂v/∂y] = 0

VELOCITY AND ACCELERATION

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Let V= Resultant velocity at any point in a fluid flow. (u,v,w) are the velocity components in x,y and z directions which are functions of space coordinates and time. www.getmyuun=iu.(cx,oym,z,t) ; v=v(x,y,z,t) ;w=w(x,y,z,t). Resultant velocity=V=ui+vj+wk |V|= (u2+v2+w2)1/2 Let ax,ay,az are the total accelerations in the x,y,z directions respectively ax= [du/ dt]= [∂u/ ∂x] [∂x/ ∂t]+ [∂u/ ∂y][ ∂y/ ∂t]+ [∂u/ ∂z][ ∂z/ ∂t]+ [∂u/ ∂t] ax= [du/dt] =u[∂u/ ∂x]+v[∂u/ ∂y]+w[∂u/ ∂z]+[ ∂u/∂ t]

Similarly, ay= [dv/ dt] = u[∂v/ ∂x]+ v[∂v/ ∂y]+w[∂v/ ∂z]+[ ∂v/∂t] az= [dw/dt] = u[∂w/ ∂x]+ v[∂w/ ∂y]+w[∂w/ ∂z]+[ ∂w/ ∂t]

1. Convective Acceleration Terms – The first three terms in the expressions for ax, ay, az. Convective acceleration is defined as the rate of change of velocity due to change of position of the fluid particles in a flow field

2. Local Acceleration Terms- The 4th term, [∂ ( )/ ∂t] in the expressions for ax, ay, az. Local or temporal acceleration is the rate of change of velocity with respect to time at a given point in a flow field.

Material or Substantial Acceleration = Convective Acceleration + Local or Temporal Acceleration. In a steady flow, temporal or local acceleration is zero. In uniform flow, convective acceleration is zero.

For steady flow, [∂u/ ∂t]= [∂v/ ∂t]= [∂w/ ∂t]= 0 ax= [du/dt]= u[∂u/∂x]+v[∂u/∂y]+w[∂u/∂z] ay= [dv/dt]= u[∂v/∂x]+v[∂v/∂y]+w[∂v/∂z] az= [dw/dt]= u[∂w/∂x]+v[∂w/∂y]+w[∂w/∂z] Acceleration Vector=axi+ayj+azk; |A|=[ax2+aY2+az2]1/2

Problem 1 The fluid flow field is given by V=x2yi+y2zj-(2xyz+yz2)k . Prove that this is a case of a possible steady incompressible flow field. u=x2y; v=y2z; w= -2xyz -yz2 (∂u/∂x) = 2xy; (∂v/∂y)=2yz; (∂w/∂z)= -2xy -2yz For steady incompressible flow, the continuity equation is [∂u/∂x + ∂v/∂y + ∂w/∂z] =0 2xy + 2yz -2xy -2yz =0 Therefore, the given flow field is a possible case of steady incompressible fluid flow.

Problem-2.

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Given v=2y2 and w=2xyz, the two velocity components. Determine the third component such that it satisfies the continuity equation. v=2y2; w=2xyz; (∂v/∂y)=4y; (∂w/∂z)=2xy www.getmyu(n∂iu./∂cxo) m+ (∂v/∂y) + (∂w/∂z) =0 (∂u/∂x) = -4y -2xy; ∂u = (-4y -2xy) ∂x u= -4xy -x2y+ f(y,z); f(y,z) can not be the function of (x)

Problem-3. Find the acceleration components at a point (1,1,1) for the following flow field: u=2x2+3y; v= -2xy+3y2+3zy; w= -(3/2)z2+2xz -9y2z ax= [∂u/∂t]+ u[∂u/∂x]+v[∂u/∂y]+w[∂u/∂z] 0+(2x2+3y)4x+(-2xy+3y2+3zy)3+0 ; [ax] 1,1,1= 32 units Similarly, ay=[∂v/∂t]+ u[∂v/∂x]+v[∂v/∂y]+w[∂v/∂z] ay = 0+(2x2+3y) (-2x)+ (-2xy+3y2+3zy)(-2x+6y+3z)+ {-(3/2)z2+2xz -9y2z}3y [ay]1,1,1 = -7.5 units Similarly, az= [∂w/∂t]+ u[∂w/∂x]+v[∂w/∂y]+w[∂w/ ∂z] Substituting, az= 23 units Resultant |a|= (ax2+ay2+az2)1/2 Problem-4. Given the velocity field V= (4+xy+2t)i + 6x3j + (3xt2+z)k. Find acceleration of a fluid particle at (2,4,-4) at t=3. [dV/dt]=[ ∂V/∂t]+u[∂V/∂x]+v[∂V/∂y]+w[∂V/∂z] u= (4+xy+2t); v=6x3; w= (3xt2+z) [∂V/∂x]= (yi+18x2j+3t2k); [∂V/∂y]= xi; [∂V/∂z]= k; [∂V/∂t] = 2i+6xtk . Substituting, [dV/dt]= (2+4y+xy2+2ty+6x4)i + (72x2+18x3y+36tx2)j + (6xt+12t2+3xyt2+6t3+z+3xt2)k The acceleration vector at the point (2,4,-4) and time t=3 is obtained by substitution, a= 170i+1296j+572k; Therefore, ax=170, ay=1296, az=572 Resultant |a|= [1702+12962+5722]1/2 units = 1426.8 units.

VELOCITY POTENTIAL AND STREAM FUNCTION

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Velocity Potential Function is a Scalar Function of space and time co-ordinates such that its negative derivatives with respect to any direction give the fluid velocity in that direction. www.getmyuΦni=.cΦo(mx,y,z) for steady flow. u= -(∂Φ/∂x); v= -(∂Φ/∂y); w= -(∂Φ/∂z) where u,v,w are the components of velocity in x,y and z directions. In cylindrical co-ordinates, the velocity potential function is given by ur= (∂Φ/∂r), uθ = (1/r)( ∂Φ/∂θ) The continuity equation for an incompressible flow in steady state is (∂u/∂x + ∂v/∂y + ∂w/∂z) = 0 Substituting for u, v and w and simplifying, (∂2Φ /∂x2 + ∂2Φ /∂y2 + ∂2Φ/∂z2) = 0 Which is a Laplace Equation. For 2-D Flow, (∂2Φ /∂x2 + ∂2Φ /∂y2) =0 If any function satisfies Laplace equation, it corresponds to some case of steady incompressible fluid flow.

IRROTATIONAL FLOW AND VELOCITY POTENTIAL

Assumption of Ir-rotational flow leads to the existence of velocity potential. Consider the rotation of the fluid particle about an axis parallel to z-axis. The rotation component is defined as the average angular velocity of two infinitesimal linear segments that are mutually perpendicular to each other and to the axis of rotation. Consider two-line segments δx, δy. The particle at P(x,y) has velocity components u,v in the x-y plane.

Fig. Rotation of a fluid partical.

The angular velocities of δx and δy are sought. The angular velocity of (δx) is {[v+ (∂v/∂x) δx –v] / δx} = (∂v/∂x) rad/sec

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