ch05 - Problem 5.1 [1] Problem 5.2 Given: Velocity fields...

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Unformatted text preview: Problem 5.1 [1] Problem 5.2 Given: Velocity fields Find: [2] Which are 3D incompressible Solution: ∂ Basic equation: ∂x u+ ∂ ∂y v+ ∂ ∂z w=0 Assumption: Incompressible flow 2 2 u ( x , y , z , t) = y + 2⋅ x⋅ z v ( x , y , z , t ) = −2⋅ y⋅ z + x ⋅ y⋅ z w ( x , y , z , t) = ∂ a) ∂ ∂ ∂x u ( x , y , z , t) → 2⋅ z ∂y ∂ Hence ∂x 2 v ( x , y , z , t ) → x ⋅ z − 2⋅ z u+ ∂ ∂y v+ ∂ ∂z w=0 u ( x , y , z , t) = x⋅ y⋅ z⋅ t v ( x , y , z , t ) = −x⋅ y⋅ z⋅ t ∂ b) ∂ ∂x u ( x , y , z , t) → t⋅ y⋅ z ∂y ∂ Hence c) ∂x 2 2 2 2 v ( x , y , z , t ) → −t ⋅ x⋅ z u+ ∂ ∂y v+ ∂ ∂z w=0 ∂z 1 22 34 ⋅x ⋅z + x ⋅y 2 2 w ( x , y , z , t) → x ⋅ z INCOMPRESSIBLE 2 w( x , y , z , t) = ∂ ∂z ( z 2 ⋅ x⋅ t − y⋅ t 2 (2 ) ) w( x , y , z , t ) → z⋅ t ⋅ x − t ⋅ y INCOMPRESSIBLE 2 u ( x , y , z , t) = x + y + z v ( x , y , z , t) = x − y + z w ( x , y , z , t) = −2⋅ x⋅ z + y + z ∂ ∂ ∂ ∂x u ( x , y , z , t) → 2⋅ x Hence ∂y ∂ ∂x v ( x , y , z , t) → −1 u+ ∂ ∂y v+ ∂ ∂z w=0 ∂z w ( x , y , z , t) → 1 − 2⋅ x INCOMPRESSIBLE Problem 5.3 [1] Problem 5.4 [2] Given: x component of velocity Find: y component for incompressible flow; Valid for unsteady?; How many y components? Solution: Basic equation: ∂ ∂x ( ρ ⋅ u) + ∂ ∂y ( ρ ⋅ v) + ∂ ∂z ( ρ⋅ w ) + ∂ ∂t ρ=0 Assumption: Incompressible flow; flow in x-y plane Hence ∂ ∂x u+ ∂ ∂y v=0 ∂ or ∂y v=− ∂ ∂x u=− ∂ ∂x [ A ⋅ x⋅ ( y − B ) ] = − A ⋅ ( y − B ) ⌠ ⎛ y2 ⎞ ⎮ v ( x , y) = −⎮ A⋅ ( y − B) dy = −A⋅ ⎜ − B⋅ y⎟ + f ( x) ⎝2 ⎠ ⌡ This basic equation is valid for steady and unsteady flow (t is not explicit) Integrating There are an infinite number of solutions, since f(x) can be any function of x. The simplest is f(x) = 0 ⎛ y2 v ( x , y) = −A⋅ ⎜ ⎝2 ⎞ − B⋅ y⎟ ⎠ 2 v ( x , y) = 6⋅ y − y 2 Problem 5.5 [2] Given: x component of velocity Find: y component for incompressible flow; Valid for unsteady? How many y components? Solution: Basic equation: ∂ ∂x ( ρ ⋅ u) + ∂ ∂y ( ρ⋅ v ) + ∂ ∂z ( ρ⋅ w) + ∂ ∂t ρ=0 Assumption: Incompressible flow; flow in x-y plane Hence ∂ ∂x u+ ∂ ∂y v=0 ( ∂ or ∂y v =− ∂ ∂x u=− ∂ ∂x ( x3 − 3⋅ x⋅ y2) = −( 3⋅ x2 − 3⋅ y2) ) ⌠ 2 2 2 3 ⎮ v ( x , y) = −⎮ 3⋅ x − 3⋅ y dy = −3⋅ x ⋅ y + y + f ( x) ⌡ This basic equation is valid for steady and unsteady flow (t is not explicit) Integrating 3 2 There are an infinite number of solutions, since f(x) can be any function of x. The simplest is f(x) = 0 v ( x , y) = y − 3⋅ x ⋅ y Problem 5.6 [2] Problem 5.7 [2] Given: y component of velocity Find: x component for incompressible flow; Simplest x components? Solution: Basic equation: ∂ ∂x ( ρ ⋅ u) + ∂ ∂y ( ρ⋅ v ) + ∂ ∂z ( ρ⋅ w) + ∂ ∂t ρ=0 Assumption: Incompressible flow; flow in x-y plane Hence Integrating ∂ ∂x u+ ∂ ∂y v=0 ∂ or ( ∂x u=− ∂ ∂y v =− ( ) ) ⌠ 3 2 3 22 1 4 ⎮ u ( x , y) = −⎮ A⋅ 3⋅ x⋅ y − x dx = − ⋅ A⋅ x ⋅ y + ⋅ A⋅ x + f ( y) 2 4 ⌡ This basic equation is valid for steady and unsteady flow (t is not explicit) There are an infinite number of solutions, since f(y) can be any function of y. The simplest is f(y) = 0 u ( x , y) = 1 43 22 ⋅ A⋅ x − ⋅ A⋅ x ⋅ y 4 2 u ( x , y) = ( ) 2 2 2 2 ∂⎡ ⎣A⋅ x⋅ y⋅ y − x ⎤ = −⎡A⋅ x⋅ y − x + A⋅ x⋅ y⋅ 2⋅ y⎤ ⎦ ⎣ ⎦ ∂y 14 22 ⋅ x − 3⋅ x y 2 Problem 5.8 [2] Given: x component of velocity Find: y component for incompressible flow; Valid for unsteady? How many y components? Solution: Basic equation: ∂ ∂x ( ρ ⋅ u) + ∂ ∂y ( ρ ⋅ v) + ∂ ∂z ( ρ⋅ w ) + Assumption: Incompressible flow; flow in x-y plane Hence Integrating ∂ ∂x u+ ∂ ∂y v=0 or ∂ ∂t ρ=0 x ⎛ ⎞ ⎞ ⎛ y ⎞ ⎟ = −⎜ A ⋅ e b ⋅ cos ⎛ y ⎞ ⎟ v = − u = − ⎜ A⋅ e ⋅ cos ⎜ ⎟ ⎟ ⎜ ⎜ ⎟⎟ ∂y ∂x ∂x ⎝ ⎝ b ⎠⎠ ⎝b ⎝ b ⎠⎠ ∂ ⎛ ∂⎜ ∂ x b ⌠ ⎮ x x ⎮Ab ⎛ y ⎞ dy = −A⋅ e b ⋅ sin ⎛ y ⎞ + f ( x) v ( x , y) = −⎮ ⋅ e ⋅ cos ⎜ ⎟ ⎜⎟ ⎝ b⎠ ⎝ b⎠ ⎮b ⌡ This basic equation is valid for steady and unsteady flow (t is not explicit) There are an infinite number of solutions, since f(x) can be any function of x. The simplest is f(x) = 0 x b y⎞ v ( x , y) = −A⋅ e ⋅ sin ⎛ ⎟ ⎜ ⎝ b⎠ x 5 y v ( x , y) = −10⋅ e ⋅ sin⎛ ⎞ ⎜⎟ ⎝ 5⎠ Problem 5.9 [3] Given: y component of velocity Find: x component for incompressible flow; Simplest x component Solution: Basic equation: ∂ ∂x ( ρ ⋅ u) + ∂ ( ρ ⋅ v) + ∂y ∂ ∂z ( ρ⋅ w ) + ∂ ∂t ρ=0 Assumption: Incompressible flow; flow in x-y plane Hence Integrating ∂ ∂x u+ ∂ ∂y v=0 2 2 ∂y v=− ⎡ 2⋅ x⋅ (x2 − 3⋅ y2)⎤ ⎤ ⎥ ⎥ = −⎢ 2 ⎢ ( 2 2)3 ⎥ ) ⎥ ⎣ x +y ⎦ ⎦ ∂ ⎡ 2⋅ x⋅ y ⎢ ∂y ⎢ 2 2 ⎣ x +y ( 2⋅ y − 2 ) 2 (x (x2 + y2) + f ( y) 2 2 2 +y 2 1 u ( x , y) = ∂ 2 1 x +y The simplest form is ∂x u=− 2 2 2 2 2 ⎡ 2⋅ x⋅ (x2 − 3⋅ y2)⎤ ⎢ ⎥ dx = x − y + f ( y) = x + y − 2⋅ y + f ( y) ⎢ ( 2 2)3 ⎥ (x2 + y2)2 (x2 + y2)2 ⎣ x +y ⎦ ⌠ ⎮ u ( x , y) = −⎮ ⎮ ⎮ ⌡ u ( x , y) = ∂ or 2 2⋅ y − x +y Note: Instead of this approach we could have verified that u and v satisfy continuity ∂⎡ ⎢ 1 ∂x ⎢ x2 + y2 ⎣ ⎤ ⎤ ⎥ + ∂ ⎡ 2⋅ x⋅ y ⎥ → 0 ⎢ 2⎥ ∂y 2 (x2 + y2) ⎦ ⎢(x2 + y2) ⎥ ⎣ ⎦ 2 − 2⋅ y However, this does not verify the solution is the simplest Problem 5.10 [2] Problem 5.11 [3] Problem 5.12 [3] Problem 5.13 [3] Given: Data on boundary layer Find: y component of velocity ratio; location of maximum value; plot velocity profiles; evaluate at particular point Solution: ⎡ 3 ⎛ y ⎞ 1 ⎛ y ⎞ 3⎤ u ( x , y) = U⋅ ⎢ ⋅ ⎜ ⎟ − ⋅⎜ ⎟⎥ ⎣ 2 ⎝ δ ( x) ⎠ 2 ⎝ δ ( x) ⎠ ⎦ so For incompressible flow Hence so and δ ( x) = c⋅ x and du 3 ⎛y = ⋅ U⋅ ⎜ − 5 dx 4⎜ 3 ⎡3 y ⎞ 1⎛ y ⎞⎤ u ( x , y) = U⋅ ⎢ ⋅ ⎛ − ⋅⎜ ⎜ ⎟ ⎟⎥ ⎣ 2 ⎝ c⋅ x ⎠ 2 ⎝ c⋅ x ⎠ ⎦ ∂ ∂x u+ ∂ ∂y v=0 ⌠ ⎮d v ( x , y) = −⎮ u ( x , y) dy ⎮ dx ⌡ 3 ⎜32 ⎝ c ⋅x 3⎟ 2⎟ c⋅ x ⎠ ⌠ ⎛ y3 x5 y x3 ⎞ ⎮3 v ( x , y) = −⎮ ⋅ U⋅ ⎜ ⋅ − ⋅ ⎟ dy ⎜ c3 2 c 2 ⎟ ⎮4 ⎝ ⎠ ⌡ 4⎞ ⎛ y2 y ⎜ ⎟ v ( x , y) = ⋅ U⋅ − 3 5⎟ 8⎜ ⎜2 3 2⎟ 2⋅ c ⋅ x ⎠ ⎝ c⋅ x 3 The maximum occurs at y⎞ ⎟ y=δ vmax = v ( x , y) = as seen in the corresponding Excel workbook δ 1⎞ ⋅ U⋅ ⋅ ⎛ 1 − ⋅ 1⎟ ⎜ 8 x⎝ 2⎠ 3 At δ = 5⋅ mm and x = 0.5⋅ m, the maximum vertical velocity is vmax U = 0.00188 2 4 1 y⎞ ⎤ δ ⎡ y⎞ ⋅ U⋅ ⋅ ⎢⎛ ⎟ − ⋅ ⎛ ⎟ ⎥ ⎜ ⎜ 2 ⎝ δ⎠ ⎦ 8 x ⎣⎝ δ ⎠ 3 To find when v /U is maximum, use Solver y /d 0.00188 1.0 v /U y /d 0.000000 0.000037 0.000147 0.000322 0.000552 0.00082 0.00111 0.00139 0.00163 0.00181 0.00188 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Vertical Velocity Distribution In Boundary layer 1.0 0.8 y /δ v /U 0.6 0.4 0.2 0.0 0.0000 0.0005 0.0010 v /U 0.0015 0.0020 Problem 5.14 [3] Problem 5.15 [3] Problem 5.16 [4] Problem 5.17 Consider a water stream from a jet of an oscillating lawn sprinkler. corresponding pathline and streakline. [5] Describe the Open-Ended Problem Statement: Consider a water stream from a jet of an oscillating lawn sprinkler. Describe the corresponding pathline and streakline. Discussion: Refer back to the discussion of streamlines, pathlines, and streaklines in Section 2-2. Because the sprinkler jet oscillates, this is an unsteady flow. Therefore pathlines and streaklines need not coincide. A pathline is a line tracing the path of an individual fluid particle. The path of each particle is determined by the jet angle and the speed at which the particle leaves the jet. Once a particle leaves the jet it is subject to gravity and drag forces. If aerodynamic drag were negligible, the path of each particle would be parabolic. The horizontal speed of the particle would remain constant throughout its trajectory. The vertical speed would be slowed by gravity until reaching peak height, and then it would become increasingly negative until the particle strikes the ground. The effect of aerodynamic drag is to reduce the particle speed. With drag the particle will not rise as high vertically nor travel as far horizontally. At each instant the particle trajectory will be lower and closer to the jet compared to the no-friction case. The trajectory after the particle reaches its peak height will be steeper than in the no-friction case. A streamline is a line drawn in the flow that is tangent everywhere to the velocity vectors of the fluid motion. It is difficult to visualize the streamlines for an unsteady flow field because they move laterally. However, the streamline pattern may be drawn at an instant. A streakline is the locus of the present locations of fluid particles that passed a reference point at previous times. As an example, choose the exit of a jet as the reference point. Imagine marking particles that pass the jet exit at a given instant and at uniform time intervals later. The first particle will travel farthest from the jet exit and on the lowest trajectory; the last particle will be located right at the jet exit. The curve joining the present positions of the particles will resemble a spiral whose radius increases with distance from the jet opening. Problem 5.18 [2] Problem 5.19 [3] Given: r component of velocity Find: θ component for incompressible flow; How many θ components Solution: Basic equation: 1∂ 1∂ ∂ ∂ ⋅ ρ⋅ r⋅ Vr + ⋅ ρ⋅ V z + ρ = 0 ρ⋅ V θ + r ∂r r ∂θ ∂z ∂t ( ) ( ) ( ) Assumption: Incompressible flow; flow in r-θ plane Hence Integrating 1∂ 1∂ ⋅ r⋅ Vr + ⋅ V =0 r ∂r r ∂θ θ () () or ⌠ Λ⋅ sin ( θ) ⎮ Λ⋅ cos ( θ) dθ = − Vθ ( r , θ) = −⎮ + f ( r) 2 2 r r ⎮ ⌡ Vθ ( r , θ) = − Λ⋅ sin ( θ) 2 + f ( r) r There are an infinite number of solutions as f(r) can be any function of r The simplest form is Vθ ( r , θ) = − Λ⋅ sin ( θ) 2 r ∂ ∂θ Vθ = − ⎜ (r⋅ Vr) = −∂r ⎛− ∂r ∂⎝ ∂ Λ⋅ cos ( θ) ⎞ r ⎟=− ⎠ Λ⋅ cos ( θ) 2 r Problem 5.20 [2] Problem 5.21 [4] 169 5.2c. (3.19) (Page 169) 5.2c. Problem 5.22 [3] Given: The velocity field Find: Whether or not it is a incompressible flow; sketch various streamlines Solution: A r Vr = Vθ = B r ( ) 1d ⋅ r ⋅ Vr = 0 r dr 1d 1d ⋅ r ⋅ Vr + ⋅ Vθ = 0 r dr r dθ ( ) Flow is incompressible For the streamlines dr r⋅ dθ = Vr Vθ r ⋅ dr r ⋅ dθ = A B so ⌠ ⌠ ⎮ 1 dr = ⎮ A dθ ⎮r ⎮B ⌡ ⌡ Integrating For incompressible flow 1d 1d ⋅ r ⋅ Vr + ⋅ Vθ = 0 r dr r dθ Hence Equation of streamlines is r = C⋅ e ( ) 1d ⋅ V =0 r dθ θ 2 ln ( r ) = A ⋅θ B A ⋅ θ + const B 4 (a) For A = B = 1 m2/s, passing through point (1m, π/2) θ− r=e 2 π 2 (b) For A = 1 m2/s, B = 0 m2/s, passing through point (1m, π/2) θ= π −4 −2 0 2 (c) For A = 0 m2/s, B = 1 m2/s, passing through point (1m, π/2) −2 r = 1⋅ m −4 (a) (b) (c) 2 4 Problem *5.23 [2] Problem *5.24 Given: Velocity field Find: [3] Stream function ψ Solution: Basic equation: ∂ ∂x ( ρ ⋅ u) + ∂ ∂y ( ρ ⋅ v) + ∂ ∂z ( ρ⋅ w ) + ∂ ∂t ρ=0 u= ∂ ∂y v=− ψ ∂ ∂x ψ Assumption: Incompressible flow; flow in x-y plane Hence ∂ ∂x u+ ∂ ∂y v=0 or ∂ Hence u = y⋅ ( 2⋅ x + 1) = and v = x⋅ ( x + 1) − y = − 2 3 Comparing these f ( x) = − ∂ ∂x 2 and 2 g ( y) = y 2 3 y x 2x + x⋅ y − − 2 2 3 2 2 3 x x⎞ ∂ ⎛y ⎜ + x⋅ y2 − − ⎟ → u ( x , y) = y + 2⋅ x⋅ y 2 3⎠ ∂y ⎝ 2 v ( x , y) = − 2 ∂⎡ ⎣x⋅ ( x + 1) − y ⎤ → 0 ⎦ ∂y 3 2 ⌠ x x 2 2 ⎮ ψ ( x , y) = −⎮ ⎡x⋅ ( x + 1) − y ⎤ dx = − − + x⋅ y + g ( y) ⎣ ⎦ 3 2 ⌡ ψ 2 2 u ( x , y) = ∂x [ y⋅ ( 2x + 2) ] + 2 ⌠ 2y ⎮ ψ ( x , y) = ⎮ y⋅ ( 2⋅ x + 1) dy = x⋅ y + + f ( x) 2 ⌡ ψ x x − 3 2 The stream function is ψ ( x , y) = Checking ∂y ∂ 2 2 3 x x⎞ ∂ ⎛y ⎜ + x⋅ y2 − − ⎟ → v ( x , y) = x2 + x − y2 2 3⎠ ∂x ⎝ 2 Problem *5.25 [2] Problem *5.26 [3] Given: The velocity field Find: Whether or not it is a incompressible flow; sketch stream function Solution: A r Vr = Vθ = ( ) 1d ⋅ r ⋅ Vr = 0 r dr ( ) Flow is incompressible For incompressible flow 1d 1d ⋅ r ⋅ Vr + ⋅ Vθ = 0 r dr r dθ Hence 1d 1d ⋅ r ⋅ Vr + ⋅ Vθ = 0 r dr r dθ ∂ For the stream function ∂θ ψ = r ⋅ Vr = A ∂ B ψ = −V θ = − r ∂r Integrating Comparing, stream function is ψ ψ = A⋅ θ − B⋅ ln ( r ) ( B r ) ψ = A⋅ θ + f ( r ) ψ = −B⋅ ln ( r ) + g( θ) 1d ⋅ V =0 r dθ θ Problem *5.27 [3] Given: Velocity field Find: Whether it's 1D, 2D or 3D flow; Incompressible or not; Stream function ψ Solution: Basic equation: ∂ ∂x ( ρ ⋅ u) + ∂ ∂y ( ρ ⋅ v) + ∂ ∂z ( ρ⋅ w ) + ∂ ∂t ρ=0 v= ∂ ∂z w=− ψ ∂ ∂y ψ Assumption: Incompressible flow; flow in y-z plane (u = 0) Velocity field is a function of y and z only, so is 2D Check for incompressible ∂ ∂y ∂ v+ ∂z w=0 ( ) 2 2 2 2 ∂⎡ ⎣y⋅ y − 3⋅ z ⎤ → 3⋅ y − 3⋅ z ⎦ ∂y Hence ∂ ∂y v+ ∂ ∂z w=0 ( Flow is INCOMPRESSIBLE (2 2 )=∂ (2 2 Hence v = y⋅ y − 3⋅ z and w = z⋅ z − 3⋅ y Comparing these f ( y) = 0 The stream function is ψ ( y , z) = z⋅ y − z ⋅ y Checking u ( y , z) = ( ∂z ) = −∂ ∂y ∂ ∂z w ( y , z) = − ( ∂y ) ⌠ 2 2 3 3 ⎮ ψ ( y , z) = −⎮ ⎡z⋅ z − 3⋅ y ⎤ dy = −y⋅ z + z⋅ y + g ( z) ⎣ ⎦ ⌡ ψ g ( z) = 0 3 (z⋅ y3 − z3⋅ y) → u (y , z) = y3 − 3⋅ y⋅ z2 ∂ ) ⌠ 2 2 3 3 ⎮ ψ ( y , z) = ⎮ y⋅ y − 3⋅ z dz = y ⋅ z − y⋅ z + f ( y) ⌡ ψ and 3 ) 2 2 2 ∂⎡ 2 ⎣z⋅ z − 3⋅ y ⎤ → 3⋅ z − 3⋅ y ⎦ ∂z (z⋅ y3 − z3⋅ y) → w (y , z) = z3 − 3⋅ y2⋅ z Problem *5.28 [3] Problem *5.29 [3] U h y x Given: Linear velocity profile Find: Stream function ψ; y coordinate for half of flow Solution: Basic equations: u= ∂ ∂y v=− ψ ∂ ∂x ψ y⎞ and we have u = U⋅ ⎛ ⎟ ⎜ ⎝ h⎠ v=0 Assumption: Incompressible flow; flow in x-y plane Check for incompressible ∂ ∂x u+ ∂ ∂y v=0 ∂ ⎛ y⎞ ⎜ U⋅ ⎟ → 0 ∂x ⎝ h ⎠ Hence ∂ ∂x u+ ∂ ∂y ∂ ∂y v=0 0→0 Flow is INCOMPRESSIBLE 2 ⌠ U⋅ y y ψ ( x , y) = ⎮ U⋅ dy = + f ( x) ⎮ 2⋅ h h ⌡ y∂ = ψ h ∂y Hence u = U⋅ and v=0=− Comparing these f ( x) = 0 The stream function is U⋅ y ψ ( x , y) = 2⋅ h For the flow (0 < y < h) ⌠ U⌠ U⋅ h Q = ⎮ u dy = ⋅ ⎮ y dy = ⌡0 ⌡0 h 2 For half the flow rate Q⌠ =⎮ ⌡0 2 Hence hhalf = ∂ ∂x ⌠ ⎮ ψ ( x , y) = −⎮ 0 dx = g ( y) ⌡ ψ 2 g ( y) = and U⋅ y 2⋅ h 2 h h hhalf 2 hhalf U⌠ u dy = ⋅ ⎮ h ⌡0 12 ⋅h 2 y dy = U⋅ hhalf 2⋅ h 2 = 1 ⎛ U⋅ h ⎞ U⋅ h ⋅⎜ ⎟= 2⎝ 2 ⎠ 4 hhalf = 1 2 ⋅h = 1.5⋅ m 2⋅ s = 1.06⋅ m s Problem *5.30 [3] Problem *5.31 [3] Problem *5.32 [3] Problem *5.33 [3] Given: Data on boundary layer Find: Stream function; locate streamlines at 1/4 and 1/2 of total flow rate Solution: ⎡ 3 ⎛ y ⎞ 1 ⎛ y ⎞ 3⎤ u ( x , y) = U⋅ ⎢ ⋅ ⎜ ⎟ − ⋅ ⎜ ⎟ ⎥ ⎣2 ⎝ δ ⎠ 2 ⎝ δ ⎠ ⎦ For the stream function u = Hence δ ( x) = c⋅ x and ⎡ 3 ⎛ y ⎞ 1 ⎛ y ⎞ 3⎤ ψ = U⋅ ⎢ ⋅ ⎜ ⎟ − ⋅ ⎜ ⎟ ⎥ ∂y ⎣2 ⎝ δ ⎠ 2 ⎝ δ ⎠ ⎦ ∂ ⌠ ⎮ ⎡ 3 y ⎞ 1 y ⎞ 3⎤ ψ = ⎮ U⋅ ⎢ ⋅ ⎛ ⎟ − ⋅ ⎛ ⎟ ⎥ dy ⎜ ⎜ ⎮ ⎣2 ⎝ δ ⎠ 2 ⎝ δ ⎠ ⎦ ⌡ ⎛3 y 1 y ⎞ ψ = U⋅ ⎜ ⋅ − ⋅ ⎟ + f ( x) ⎜ 4 δ 8 δ3 ⎟ ⎝ ⎠ 2 4 ⎡ 3 y 2 1 y 4⎤ ψ = U⋅ δ⋅ ⎢ ⋅ ⎛ ⎞ − ⋅ ⎛ ⎞ ⎥ ⎜⎟ ⎜⎟ ⎣4 ⎝ δ ⎠ 8 ⎝ δ ⎠ ⎦ Let ψ = 0 = 0 along y = 0, so f(x) = 0, so The total flow rate in the boundary layer is Q 31 5 = ψ ( δ) − ψ ( 0) = U⋅ δ⋅ ⎛ − ⎞ = ⋅ U⋅ δ ⎜ ⎟ W ⎝ 4 8⎠ 8 At 1/4 of the total ⎡ 3 y ⎞ 2 1 y ⎞ 4⎤ 1 5 ⎞ ψ − ψ0 = U⋅ δ⋅ ⎢ ⋅ ⎛ ⎟ − ⋅ ⎛ ⎟ ⎥ = ⋅ ⎛ ⋅ U⋅ δ⎟ ⎜ ⎜ ⎜ ⎣4 ⎝ δ ⎠ 8 ⎝ δ ⎠ ⎦ 4 ⎝ 8 ⎠ 2 4 y⎞ y⎞ 24⋅ ⎛ ⎟ − 4⋅ ⎛ ⎟ = 5 ⎜ ⎜ δ⎠ ⎝ ⎝ δ⎠ The solution to the quadratic is X = Hence y = δ or 2 4⋅ X − 24⋅ X + 5 = 0 where 2 24 − 24 − 4⋅ 4⋅ 5 2⋅ 4 X = 0.216 Note that the other root is 2 X= 24 + y δ 2 24 − 4⋅ 4⋅ 5 = 5.784 2⋅ 4 X = 0.465 ⎡ 3 y ⎞ 2 1 y ⎞ 4⎤ 1 5 ⎞ At 1/2 of the total flow ψ − ψ0 = U⋅ δ⋅ ⎢ ⋅ ⎛ ⎟ − ⋅ ⎛ ⎟ ⎥ = ⋅ ⎛ ⋅ U⋅ δ⎟ ⎜ ⎜ ⎜ ⎣4 ⎝ δ ⎠ 8 ⎝ δ ⎠ ⎦ 2 ⎝ 8 ⎠ 2 4 y⎞ y⎞ 12⋅ ⎛ ⎟ − 2⋅ ⎛ ⎟ = 5 ⎜ ⎜ ⎝ δ⎠ ⎝ δ⎠ The solution to the quadratic is X = Hence y = δ 12 − X = 0.671 or 2 2⋅ X − 12⋅ X + 5 = 0 where 2 12 − 4⋅ 2⋅ 5. 2⋅ 2 X = 0.450 Note that the other root is 2 X= 12 + y δ 2 12 − 4⋅ 2⋅ 5 = 5.55 2⋅ 2 Problem *5.34 [3] Problem *5.35 [3] Problem 5.36 [3] Given: Velocity field Find: Whether flow is incompressible; Acceleration of particle at (2,1) Solution: ∂ Basic equations ∂x ∂ u+ ∂y v=0 (4 22 ∂ For incompressible flow ∂x ∂ u+ ∂y ) ( 4 u ( x , y) = A⋅ x − 6⋅ x ⋅ y + y ( ) ( ) 4 22 4 3 2 ∂⎡ ⎣A⋅ x − 6⋅ x ⋅ y + y ⎤ → A⋅ 4⋅ x − 12⋅ x⋅ y ⎦ ∂x Hence ∂ ∂x ∂ ∂y 3 ) v=0 Checking u+ 3 v ( x , y) = A⋅ 4⋅ x⋅ y − 4⋅ x ⋅ y ( ) ( ) 3 3 3 2 ∂⎡ ⎣A⋅ 4⋅ x⋅ y − 4⋅ x ⋅ y ⎤ → −A⋅ 4⋅ x − 12⋅ x⋅ y ⎦ ∂y v=0 The acceleration is given by For this flow ax = u⋅ ∂ ∂x u + v⋅ ∂ ∂y (4 u ) 4∂ 22 ax = A⋅ x − 6⋅ x ⋅ y + y ⋅ 2 (2 ∂ ∂x v + v⋅ ∂ ∂y (4 )3 v ) 4∂ 22 ay = A⋅ x − 6⋅ x ⋅ y + y ⋅ 2 (2 ) 2 ay = 4⋅ A ⋅ y⋅ x + y 2 Hence at (2,1) ∂y 2 ax = 4⋅ A ⋅ x⋅ x + y ay = u⋅ ∂x ⎡A⋅ (x4 − 6⋅ x2⋅ y2 + y4)⎤ + A⋅ (4⋅ x⋅ y3 − 4⋅ x3⋅ y)⋅ ∂ ⎡A⋅ (x4 − 6⋅ x2⋅ y2 + y4)⎤ ⎣ ⎦ ⎣ ⎦ ∂x ⎡A⋅ (4⋅ x⋅ y3 − 4⋅ x3⋅ y)⎤ + A⋅ (4⋅ x⋅ y3 − 4⋅ x3⋅ y)⋅ ∂ ⎡A⋅ (4⋅ x⋅ y3 − 4⋅ x3⋅ y)⎤ ⎣ ⎦ ⎣ ⎦ ∂y 3 3 m ax = 62.5 2 s 3 1 1⎞ 2 2 ay = 4 × ⎛ ⋅ × 1⋅ m × ⎡( 2⋅ m) + ( 1⋅ m) ⎤ ⎣ ⎦ ⎜4 3 ⎟ ⎝ m ⋅s ⎠ m ay = 31.3 2 s 1 1⎞ 2 2⎤ ⎡ ax = 4 × ⎛ ⋅ ⎜ 4 3 ⎟ × 2⋅ m × ⎣( 2⋅ m) + ( 1⋅ m) ⎦ ⎝ m ⋅s ⎠ 2 a= 2 ax + ay 2 a = 69.9 m 2 s Problem 5.37 [2] Problem 5.38 [2] Problem 5.39 [2] Problem 5.40 [3] Given: x component of velocity field Find: Simplest y component for incompressible flow; Acceleration of particle at (1,3) Solution: ∂ ∂ Basic equations u= We are given u ( x , y) = A⋅ x − 10⋅ x ⋅ y + 5⋅ x⋅ y ∂y v=− ψ (5 ∂x ψ 32 4 ) ( ) ⌠ ⌠ 10 3 3 5 32 4 5⎞ ⎮ ⎮ ⎛5 Hence for incompressible flow ψ ( x , y) = ⎮ u dy = ⎮ A⋅ x − 10⋅ x ⋅ y + 5⋅ x⋅ y dy = A⋅ ⎜ x ⋅ y − 3 ⋅ x ⋅ y + x⋅ y ⎟ + f ( x) ⎝ ⎠ ⌡ ⌡ v ( x , y) = − ∂ ∂x () ψ xy = − ( 23 5) (4 + F ( x) 4 23 5 v ( x , y) = −A⋅ ( 5⋅ x ⋅ y − 10⋅ x ⋅ y + y ) v ( x , y) = −A⋅ 5⋅ x ⋅ y − 10⋅ x ⋅ y + y Hence The simplest is ) 10 3 3 5⎞ 4 23 5 ∂⎡ ⎛5 ⎤ ⎢A⋅ ⎜ x ⋅ y − ⋅ x ⋅ y + x⋅ y ⎟ + f ( x)⎥ = −A⋅ 5⋅ x ⋅ y − 10⋅ x ⋅ y + y + F ( x) 3 ⎠ ⎦ ∂x ⎣ ⎝ where F(x) is an arbitrary function of x The acceleration is given by For this flow (5 ax = u⋅ ∂ ∂x u + v⋅ ∂ ∂y u ) 4∂ 32 ax = A⋅ x − 10⋅ x ⋅ y + 5⋅ x⋅ y ⋅ 2 ∂x (2 (5 ∂ ∂x v + v⋅ ∂ ∂y ) ) ay = A⋅ x − 10⋅ x ⋅ y + 5⋅ x⋅ y ⋅ 2 ( 2 4 v 4∂ 32 ∂y 2 ax = 5⋅ A ⋅ x⋅ x + y ay = u⋅ ⎡A⋅ (x5 − 10⋅ x3⋅ y2 + 5⋅ x⋅ y4)⎤ − A⋅ (5⋅ x4⋅ y − 10⋅ x2⋅ y3 + y5)⋅ ∂ ⎡A⋅ (x5 − 10⋅ x3⋅ y2 + 5⋅ x⋅ y4)⎤ ⎣ ⎦ ⎣ ⎦ ∂x ) 2 ay = 5⋅ A ⋅ y⋅ x + y ⎡−A⋅ (5⋅ x4⋅ y − 10⋅ x2⋅ y3 + y5)⎤ − A⋅ (5⋅ x4⋅ y − 10⋅ x2⋅ y3 + y5)⋅ ∂ ⎡−A⋅ (5⋅ x4⋅ y − 10⋅ x2⋅ y3 + y5)⎤ ⎣ ⎦ ⎣ ⎦ ∂y 4 2 Hence at (1,3) 4 1 1⎞ 2 2 ax = 5 × ⎛ ⋅ × 1⋅ m × ⎡( 1⋅ m) + ( 3⋅ m) ⎤ ⎣ ⎦ ⎜2 4 ⎟ ⎝ m ⋅s ⎠ ax = 1.25 × 10 s 2 4 1 1⎞ 2 2 ay = 5 × ⎛ ⋅ × 3⋅ m × ⎡( 1⋅ m) + ( 3⋅ m) ⎤ ⎣ ⎦ ⎜2 4 ⎟ ⎝ m ⋅s ⎠ 4m 2 ay = 3.75 × 10 4m 2 s a= 2 ax + ay 2 4m 2 a = 3.95 × 10 s Problem 5.41 [2] Given: Velocity field Find: Whether flow is incompressible; expression for acceleration; evaluate acceleration along axes and along y = x Solution: 2 The given data is For incompressible flow Hence, checking A = 10⋅ ∂ ∂x ∂ ∂x m s ∂ u+ ∂y ∂ u+ ∂y A⋅ x u ( x , y) = 2 A⋅ y v ( x , y) = 2 2 x +y 2 x +y v=0 v = −A ⋅ (x2 − y2) + A⋅ (x2 − y2) = 0 (x2 + y2)2 (x2 + y2)2 Incompressible flow The acceleration is given by (2 ) du du A⋅ x ⎡ A⋅ x − y ⎤ ⎤ ⎥ + A⋅ y ⋅ ⎡− 2⋅ A⋅ x⋅ y ⎥ For the present steady, 2D flow ax = u⋅ + v⋅ = ⋅ ⎢− ⎢ 2 2⎢ 2⎥ 2 2 2 dx dy 2 2 x +y x + y ⎢ x2 + y2 ⎥ ⎣ x +y ⎦ ⎣ ⎦ 2 ( ) ( (2 ) ) dv dv A⋅ x ⎡ 2⋅ A⋅ x⋅ y ⎤ A⋅ y ⎡ A⋅ x − y ⎤ ⎥ ay = u⋅ + v⋅ = ⋅ ⎢− ⎥ + 2 2 ⋅⎢ 2 2 2 2 dx dy x + y ⎢ x2 + y2 ⎥ x + y ⎢ x2 + y2 ⎥ ⎣ ⎦ ⎣ ⎦ ( ) ( 2 ax = − A ⋅x (x2 + y2)2 2 ) 2 ay = − A ⋅y (x2 + y2) 2 2 Along the x axis A 100 ax = − =− 3 3 x x ay = 0 Along the y axis ax = 0 A 100 ay = − =− 3 3 y y Along the line x = y ax = − 2 2 A ⋅x 4 r where r= 2 =− 2 100⋅ x ay = − 4 A ⋅y 4 =− 100⋅ y r r 4 r 2 x +y For this last case the acceleration along the line x = y is 2 a= 2 A A 100 2 2 2 2 ax + ay = − ⋅ x + y = − =− 4 3 3 r r r a=− A 2 3 r =− 100 3 r In each case the acceleration vector points towards the origin, proportional to 1/distance3, so the flow field is a radial decelerating flow Problem 5.42 [2] Problem 5.43 [2] Problem 5.44 [4] Given: Flow in a pipe with variable diameter Find: Expression for particle acceleration; Plot of velocity and acceleration along centerline Solution: Assumptions: 1) Incompressible flow 2) Flow profile remains unchanged so centerline velocity can represent average velocity Basic equations Q = V⋅ A For the flow rate Q = V⋅ A = V⋅ But D = Di + Hence ⎤ ⎡ ( Do − Di) ⋅ x⎥ π⋅ ⎢Di + π⋅ D i L ⎦ Vi⋅ = V⋅ ⎣ 2 π⋅ D 4 ( Do − Di) ⋅ x where Di and Do are the inlet and exit diameters, and x is distance along the pipe of length L: D(0) = Di, D(L) = Do. L 2 2 4 4 2 V = Vi⋅ Di ⎡ (Do − Di) ⋅ x⎤ ⎢Di + ⎥ L ⎣ ⎦ Some representative values are V ( 0⋅ m) = 1 2 Vi = 2 ⎛ Do ⎞ ⎤ − 1⎟ ⎥ ⎜ Di ⎝ ⎠ ⋅ x⎥ ⎥ L ⎦ L m V ⎛ ⎞ = 2.56 ⎜⎟ s ⎝ 2⎠ ⎡ ⎢ ⎢ ⎢1 + ⎣ m s Vi V ( x) = ⎡ ⎢ ⎢ ⎢1 + ⎣ V ( L) = 16 ⎛ Do ⎞ ⎤ − 1⎟ ⎥ ⎜ Di ⎝ ⎠ ⋅ x⎥ ⎥ L ⎦ 2 m s The acceleration is given by 2 ⎛ Do For this flow ax = V⋅ ∂ ∂x V ax = Vi ⎡ ⎢ ⎢ ⎢1 + ⎣ ⎛ Do ⎞ ⎤ − 1⎟ ⎥ ⎜ Di ⎝ ⎠ ⋅ x⎥ ⎥ L ⎦ 2 ⋅ 2⋅ Vi ⋅ ⎜ ⎞ − 1⎟ ⎤ ⎝ Di ⎠ ⎥=− 2 5 ∂x ⎢ ⎡ ⎛ Do ⎞ ⎤ ⎥ ⎡ ⎛ Do ⎞ ⎤ ⎢⎢ ⎜ ⎢ x⋅ ⎜ ⎥ − 1⎟ ⎥ ⎥ − 1⎟ ⎢ ⎢ ⎝ Di ⎠ ⎥ ⎥ ⎢ ⎝ Di ⎠ ⎥ L⋅ ⎢ ⋅ x⎥ ⎥ + 1⎥ ⎢ ⎢1 + L L ⎣⎣ ⎦⎦ ⎣ ⎦ ∂⎡ ⎢ Vi 2 ⎛ Do 2⋅ V i ⋅ ⎜ ax ( x) = ⎝ Di ⎞ − 1⎟ ⎠ ⎡ ⎛ Do ⎞ ⎤ ⎢ x⋅ ⎜ ⎥ − 1⎟ ⎢ ⎝ Di ⎠ ⎥ L⋅ ⎢ + 1⎥ L ⎣ ⎦ 5 m ⎛L ax ⎜ ⎞ = −7.864 ⎟ 2 ⎝ 2⎠ s m Some representative values are ax ( 0⋅ m) = −0.75 2 s m ax ( L) = −768 2 s The following plots can be done in Excel 20 V (m/s) 15 10 5 0 0.5 1 1.5 2 1.5 2 x (m) a (m/s2) 0 0.5 1 − 200 − 400 − 600 − 800 x (m) Problem 5.45 [2] Problem 5.46 [2] Problem 5.47 [4] Given: Data on pollution concentration Find: Plot of concentration; Plot of concentration over time for moving vehicle; Location and value of maximum rate change Solution: Basic equation: Material derivative D ∂ ∂ ∂∂ =u +v +w + Dt ∂x ∂y ∂z ∂t v=0 ⎛ −x − x ⎞ ⎜a 2⋅ a ⎟ c ( x) = A⋅ ⎝ e −e ⎠ For this case we have u=U w=0 Hence Dc dc U⋅ A ⎜ 1 2⋅ a ⎟⎥ 2⋅ a a⎟ d⎢ ⎜ a = u⋅ −e ⋅⎜ ⋅e −e ⎟ = U⋅ ⎣A⋅ ⎝ e ⎠⎦ = Dt dx a ⎝2 dx ⎠ ⎡⎛ − x − x ⎞⎤ ⎛ − x − x⎞ We need to convert this to a function of time. For this motion u = U so x = U⋅ t ⎛ − U⋅ t − Dc U⋅ A ⎜ 1 2⋅ a = ⋅⎜ ⋅e −e Dt a ⎝2 U⋅ t ⎞ a⎟ ⎟ ⎠ The following plots can be done in Excel c (ppm) 0 2 4 6 −6 − 1×10 −6 − 2×10 −6 − 3×10 x (m) 8 10 −5 Dc/Dt (ppm/s) 5×10 0 0.1 0.2 0.3 0.4 0.5 −5 − 5×10 −4 − 1×10 t (s) The maximum rate of change is when ⎡ ⎛ − x − x ⎞⎤ d ⎛ Dc ⎞ d ⎢ U⋅ A ⎜ 1 2⋅ a a ⎟⎥ ⋅⎜ ⋅e − e ⎟⎥ = 0 ⎜ ⎟ = ⋅⎢ dx ⎝ Dt ⎠ dx ⎣ a ⎝ 2 ⎠⎦ ⎛ x x ⎞ − − U⋅ A ⎜ a 1 2⋅ a ⎟ ⋅ ⎜e − ⋅e ⎟=0 2 4 ⎠ a⎝ − or 1⎞ xmax = 2⋅ a⋅ ln ( 4) = 2 × 1⋅ m × ln ⎛ ⎟ ⎜ ⎝ 4⎠ tmax = xmax U = 2.77⋅ m × s 20⋅ m xmax xmax ⎞ ⎛ − ⎜ 1 − 2⋅ a ⎟ U⋅ A a = ⋅⎜ ⋅e −e ⎟ Dt a ⎝2 ⎠ 2.77 2.77 ⎞ ⎛ − Dcmax ⎜ 1 − 2⋅ 1 m 1 −5 1⎟ = 20⋅ × 10 ⋅ ppm × ×⎜ ×e −e ⎟ Dt s 1⋅ m ⎝ 2 ⎠ e x 2⋅ a = 1 4 xmax = 2.77⋅ m tmax = 0.138⋅ s Dcmax Dcmax Dt = 1.25 × 10 − 5 ppm ⋅ s Note that there is another maximum rate, at t = 0 (x = 0) Dcmax Dt = 20⋅ m 1 ⎛1 −5 ⎞ × 10 ⋅ ppm × ⋅ ⎜ − 1⎟ s 1⋅ m ⎝ 2 ⎠ Dcmax Dt − 4 ppm = −1 × 10 ⋅ s Problem 5.48 [2] Problem 5.49 [2] Problem 5.50 [3] Problem 5.51 [3] Problem 5.52 [3] Problem 5.53 [3] Problem 5.54 [3] Problem 5.55 [3] Problem 5.56 [3] Problem 5.57 [4] U y x Given: Flow in boundary layer Find: Expression for particle acceleration ax; Plot acceleration and find maximum at x = 0.8 m Solution: Basic equations u ⎛ y⎞ ⎛ y⎞ = 2⋅ ⎜ ⎟ − ⎜ ⎟ U ⎝ δ⎠ ⎝ δ⎠ We need to evaluate ax = u⋅ First, substitute λ ( x , y) = Then ∂ ∂x u + v⋅ ∂ ∂y 3 v δ ⎡1 ⎛ y ⎞ 1 ⎛ y ⎞ ⎤ = ⋅⎢ ⋅⎜ ⎟ − ⋅⎜ ⎟ ⎥ U x ⎣2 ⎝ δ ⎠ 3 ⎝ δ ⎠ ⎦ 2 δ = c⋅ x u y δ ( x) v δ1 13 = ⋅⎛ ⋅λ − ⋅λ ⎞ ⎜ ⎟ U x ⎝2 3⎠ u 2 = 2⋅ λ − λ U so − du dλ y ⎞ dδ u= ⋅ = U⋅ ( 2 − 2⋅ λ) ⋅ ⎛ − ⎟ ⋅ ⎜ 2 dx dλ dx ∂x ⎝ δ⎠ ∂ λ1 u = U⋅ ( 2 − 2⋅ λ) ⋅ ⎛ − ⎞ ⋅ ⋅ c⋅ x ⎜⎟ ∂x ⎝ δ⎠ 2 ∂ − ( 1 2 − λ ⎞1 = U⋅ ( 2 − 2⋅ λ) ⋅ ⎛ − ⋅ ⋅ c⋅ x ⎜ 1⎟ 2 ⎜ 2⎟ ⎝ c⋅ x ⎠ 1 2 ) 2 λ U⋅ λ − λ =− u = −U⋅ ( 2 − 2⋅ λ) ⋅ x 2⋅ x ∂x ∂ 1 dδ 1 2 = ⋅ c⋅ x dx 2 ( ) 2 2 y ⎞ 2⋅ U ⎡ y ⎛ y ⎞ ⎤ 2⋅ U⋅ λ − λ ⎛2 ⋅⎢ − ⎜ ⎟ ⎥ = u = U⋅ ⎜ − 2⋅ ⎟ = 2 δ y δ ⎣δ ⎝ δ ⎠ ⎦ ∂y δ⎠ ⎝ ∂ ∂ ( )( )⎤ + U⋅ δ ⋅ ⎛ 1 ⋅ λ − 1 ⋅ λ3⎞ ⋅ ⎡2⋅ U⋅ (λ − λ2)⎤ ⎥ ⎢ ⎥ 2 2 ⎡ U⋅ λ − λ u = U ⋅ 2⋅ λ − λ ⎢ ⎜ ax = u⋅ Collecting terms To find the maximum 2 dax U⎛ 243 =0= ⋅ ⎜ −2⋅ λ + 4⋅ λ − ⋅ λ ⎞ ⎟ dλ x⎝ 3⎠ ⎟ ⎠⎣ 2 2 2 3 4 U ⎛ 2 4 3 1 4⎞ U ⎡ ⎛ y⎞ 4 y⎞ 1 y⎞ ⎤ ax = ⋅ ⎜ −λ + ⋅ λ − ⋅ λ ⎟ = ⋅ ⎢−⎜ ⎟ + ⋅ ⎛ ⎟ − ⋅ ⎛ ⎟ ⎥ ⎜ ⎜ x⎝ x ⎣ ⎝ δ⎠ 3 3⎠ 3 ⎝ δ⎠ 3 ⎝ δ⎠ ⎦ The solution of this quadratic (λ < 1) is ∂x u + v⋅ ∂ Hence ∂y λ= 3− 3 2 ⎣ x ⎦ x ⎝2 3 or −1 + 2⋅ λ − λ = 0.634 y = 0.634 δ y 22 ⋅λ = 0 3 ⎦ 2 2 U⎛ U 24 31 4 ax = ⋅ ⎜ −0.634 + ⋅ 0.634 − ⋅ 0.634 ⎞ = −0.116⋅ ⎟ x⎝ x 3 3 ⎠ At λ = 0.634 2 1 ⎛ m⎞ ax = −0.116 × ⎜ 6⋅ ⎟ × s⎠ 0.8⋅ m ⎝ m ax = −5.22 2 s The following plot can be done in Excel 1 0.9 0.8 0.7 y/d 0.6 0.5 0.4 0.3 0.2 0.1 −6 −5 −4 −3 a (m/s2) −2 −1 0 Problem 5.58 [3] Part 1/2 Problem 5.58 [3] Part 2/2 Problem 5.59 [3] Problem 5.60 [3] Problem 5.61 [3] Part 1/2 Problem 5.61 [3] Part 2/2 A0 = L= b= λ= U0 = 0.5 5 0.1 0.2 5 m2 m m-1 s-1 m/s 0 5 10 60 t= 2 2 2 x (m) a x (m/s ) a x (m/s ) a x (m/s ) a x (m/s2) 0.0 1.00 1.367 2.004 2.50 0.5 1.05 1.552 2.32 2.92 1.0 1.11 1.78 2.71 3.43 1.5 1.18 2.06 3.20 4.07 2.0 1.25 2.41 3.82 4.88 2.5 1.33 2.86 4.61 5.93 3.0 1.43 3.44 5.64 7.29 3.5 1.54 4.20 7.01 9.10 4.0 1.67 5.24 8.88 11.57 4.5 1.82 6.67 11.48 15.03 5.0 2.00 8.73 15.22 20.00 For large time (> 30 s) the flow is essentially steady-state Acceleration ax (m/s2) Acceleration in a Nozzle 22 20 18 16 14 12 10 8 6 4 2 0 t=0s t=1s t=2s t = 10 s 0.0 0.5 1.0 1.5 2.0 2.5 x (m) 3.0 3.5 4.0 4.5 5.0 Problem 5.63 [3] Part 1/2 Problem 5.63 [3] Part 2/2 Problem 5.64 [4] 5.53 5.53 5.53 Problem 5.65 [4] Problem 5.66 Given: Velocity components Find: [2] Which flow fields are irrotational Solution: ∂ For a 2D field, the irrotionality the test is (a) (b) (c) (d) ∂ ∂x ∂ ∂x ∂ ∂x ∂ ∂x v− v− v− v− ∂ ∂y ∂ ∂y ∂ ∂y ∂ ∂y ∂x ( v− )( ∂ ∂y u=0 ) 2 2 2 2 2 u = ⎡3⋅ x + y − 2⋅ y ⎤ − 2⋅ y − x = 4⋅ x + y − 4⋅ y ≠ 0 ⎣ ⎦ Not irrotional u = ( 2⋅ y + 2⋅ x) − ( 2⋅ y − 2⋅ x) = 4⋅ x ≠ 0 Not irrotional ( 2) − (2) = t2 − 2 ≠ 0 u= t Not irrotional u = ( −2⋅ y⋅ t) − ( 2⋅ x⋅ t) = −2⋅ x⋅ t − 2⋅ y⋅ t ≠ 0 Not irrotional Problem 5.67 Given: Flow field Find: [3] If the flow is incompressible and irrotational Solution: ∂ Basic equations: Incompressibility a) 7 ∂x b) ∂ ∂x 34 ∂y v =0 6 6 42 24 ∂ ∂y 6 v≠0 ∂ 52 34 ∂x 5 ∂ ∂y ∂y u=0 43 25 7 ∂ 6 ∂y 42 24 6 v ( x , y) → 7⋅ x − 105⋅ x ⋅ y + 105⋅ x ⋅ y − 7⋅ y v ( x , y) = 7⋅ x ⋅ y − 35⋅ x ⋅ y + 21⋅ x ⋅ y − y 6 5 33 − u≠0 Note that if we define ∂ 6 v ( x , y) → 42⋅ x ⋅ y − 140⋅ x ⋅ y + 42⋅ x⋅ y v− v− COMPRESSIBLE 7 ∂x ∂x 6 u ( x , y) → 7⋅ x − 105⋅ x ⋅ y + 105⋅ x ⋅ y − 7⋅ y u+ ∂ Irrotationality v ( x , y) = 7⋅ x ⋅ y − 35⋅ x ⋅ y + 21⋅ x ⋅ y − y u ( x , y) = x − 21⋅ x ⋅ y + 35⋅ x ⋅ y − 7⋅ x⋅ y ∂ Hence 52 ∂ u ( x , y) = x − 21⋅ x ⋅ y + 35⋅ x ⋅ y − 7⋅ x⋅ y ∂ Hence ∂x u+ ∂ ∂y 43 5 25 33 7 5 u ( x , y) → 42⋅ x ⋅ y − 140⋅ x ⋅ y + 42⋅ x⋅ y ROTATIONAL ( 6 43 25 ) 7 v ( x , y) = − 7⋅ x ⋅ y − 35⋅ x ⋅ y + 21⋅ x ⋅ y − y then the flow is incompressible and irrotational! Problem 5.68 [2] 5.12 Problem 5.69 [2] Problem 5.70 [2] Problem *5.71 Given: Stream function Find: [3] If the flow is incompressible and irrotational Solution: Basic equations: ∂ Incompressibility u+ ∂ v =0 Irrotationality ∂ ∂x ∂y ∂x Note: The fact that ψ exists means the flow is incompressible, but we check anyway 6 42 24 v− ∂ ∂y u=0 6 ψ ( x , y) = x − 15⋅ x ⋅ y + 15⋅ x ⋅ y − y Hence u ( x , y) = ∂ ∂y 23 4 5 ψ ( x , y) → 60⋅ x ⋅ y − 30⋅ x ⋅ y − 6⋅ y v ( x , y) = − ∂ ∂x 32 5 For incompressibility ∂ ∂x Hence ∂ ∂x 3 ∂ 3 u ( x , y) → 120⋅ x⋅ y − 120⋅ x ⋅ y u+ ∂ ∂y 3 ∂y v=0 3 v ( x , y) → 120⋅ x ⋅ y − 120⋅ x⋅ y INCOMPRESSIBLE For irrotationality ∂ ∂x Hence ∂ ∂x 22 4 4 v ( x , y) → 180⋅ x ⋅ y − 30⋅ x − 30⋅ y v− ∂ ∂y u=0 − ∂ ∂y 4 ψ ( x , y) → 60⋅ x ⋅ y − 6⋅ x − 30⋅ x⋅ y 4 22 4 u ( x , y) → 30⋅ x − 180⋅ x ⋅ y + 30⋅ y IRROTATIONAL Problem *5.72 Given: Stream function Find: [3] If the flow is incompressible and irrotational Solution: Basic equations: ∂ Incompressibility u+ ∂ v =0 Irrotationality ∂ ∂x ∂y ∂x Note: The fact that ψ exists means the flow is incompressible, but we check anyway 5 33 v− ∂ ∂y u=0 5 ψ ( x , y) = 3⋅ x ⋅ y − 10⋅ x ⋅ y + 3⋅ x⋅ y Hence u ( x , y) = ∂ ∂y 5 32 4 ψ ( x , y) → 3⋅ x − 30⋅ x ⋅ y + 15⋅ x⋅ y v ( x , y) = − ∂ ∂x 23 4 For incompressibility ∂ ∂x Hence ∂ ∂x 4 22 4 u ( x , y) → 15⋅ x − 90⋅ x ⋅ y + 15⋅ y u+ ∂ ∂y v=0 ∂ 22 ∂y 4 4 v ( x , y) → 90⋅ x ⋅ y − 15⋅ x − 15⋅ y INCOMPRESSIBLE For irrotationality ∂ ∂x Hence ∂ ∂x 3 3 v ( x , y) → 60⋅ x⋅ y − 60⋅ x ⋅ y v− ∂ ∂y u=0 − ∂ ∂y 5 ψ ( x , y) → 30⋅ x ⋅ y − 15⋅ x ⋅ y − 3⋅ y 3 3 u ( x , y) → 60⋅ x ⋅ y − 60⋅ x⋅ y IRROTATIONAL Problem *5.73 [2] Given: The stream function Find: Whether or not the flow is incompressible; whether or not the flow is irrotational Solution: The stream function is A ψ=− (2 2 2⋅ π x + y The velocity components are u= dψ = dy ) A⋅ y (2 2 π x +y ) v=− 2 dψ =− dx A⋅ x (2 Because a stream function exists, the flow is: Alternatively, we can check with Incompressible ∂ ∂x ∂ ∂x For a 2D field, the irrotionality the test is )2 2 π x +y ∂ ∂x ∂ ∂x u+ u+ v− v− ∂ ∂y ∂ ∂y v =0 v =− ∂ ∂y ∂ ∂y 4⋅ A⋅ x⋅ y ( 2 2 π x +y ) 3 + 4⋅ A⋅ x⋅ y ( 2 2 π x +y ) 3 =0 Incompressible u=0 u= ( 2 2) − A⋅ (3⋅ x2 − y2) = − 2⋅ A ≠ 0 3 3 2 2 2 2 2 2 2 π⋅ ( x + y ) π⋅ ( x + y ) π⋅ ( x + y ) A⋅ x − 3⋅ y Not irrotational Problem *5.74 [2] Problem *5.75 [3] Problem *5.76 [2] Problem *5.77 [2] Problem *5.78 [2] Problem 5.79 [3] Problem *5.80 [3] Problem 5.81 [3] Problem 5.82 [2] Problem 5.83 [3] Problem 5.84 [3] Problem 5.85 [2] Problem 5.86 [2] Problem 5.87 N=4 Δx = 0.333 Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 1.333 -1.000 0.000 0.000 0.000 1.333 (RHS) 1 0 0 0 Inverse Matrix 1.000 0.750 0.563 0.422 0.000 0.750 0.563 0.422 0.000 0.000 0.750 0.563 0.000 0.000 0.000 0.750 Result 1.000 0.750 0.563 0.422 Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 x 0.000 0.333 0.667 1.000 0.000 1.333 -1.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 (RHS) 1 0 0 0 0 0 0 0 Inverse Matrix 1 1.000 0.875 0.766 0.670 0.586 0.513 0.449 0.393 2 0.000 0.875 0.766 0.670 0.586 0.513 0.449 0.393 3 0.000 0.000 0.875 0.766 0.670 0.586 0.513 0.449 4 0.000 0.000 0.000 0.875 0.766 0.670 0.586 0.513 5 0.000 0.000 0.000 0.000 0.875 0.766 0.670 0.586 6 0.000 0.000 0.000 0.000 0.000 0.875 0.766 0.670 7 0.000 0.000 0.000 0.000 0.000 0.000 0.875 0.766 8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.875 Result 1.000 0.875 0.766 0.670 0.586 0.513 0.449 0.393 Exact 1.000 0.717 0.513 0.368 Error 0.000 0.000 0.001 0.001 0.040 N=8 Δx = 0.143 x 0.000 0.143 0.286 0.429 0.571 0.714 0.857 1.000 Exact 1.000 0.867 0.751 0.651 0.565 0.490 0.424 0.368 Error 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.019 N = 16 Δx = 0.067 x 0.000 0.067 0.133 0.200 0.267 0.333 0.400 0.467 0.533 0.600 0.667 0.733 0.800 0.867 0.933 1.000 N 4 8 16 2 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 4 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 6 0.000 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 7 8 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 0.000 0.000 -1.000 1.067 0.000 0.000 -1.000 1.067 0.000 0.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 11 12 13 14 15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 16 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 (RHS) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Inverse Matrix 1.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.380 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Eq. 5.34 (LHS) 1 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.380 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 Result 1.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.380 Δx 0.333 0.143 0.067 Error 0.040 0.019 0.009 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 Exact 1.000 0.936 0.875 0.819 0.766 0.717 0.670 0.627 0.587 0.549 0.513 0.480 0.449 0.420 0.393 0.368 Error 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.009 1.0 N=4 N=8 N = 16 Exact solution 0.9 0.8 0.7 u 0.6 0.5 0.4 0.3 0.0 0.2 0.4 0.6 x 0.1 ε 0.01 Actual Error Least Squares Fit 0.001 0.01 0.10 Δx 1.00 0.8 1.0 Problem 5.88 New Eq. 5.34: − ui −1 + (1 + Δx )ui = 2Δx ⋅ sin( xi ) N=4 Δx = 0.333 Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 1.333 -1.000 0.000 0.000 0.000 1.333 (RHS) 0 0.21813 0.41225 0.56098 Inverse Matrix 1.000 0.750 0.563 0.422 0.000 0.750 0.563 0.422 0.000 0.000 0.750 0.563 0.000 0.000 0.000 0.750 Result 0.000 0.164 0.432 0.745 Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 x 0.000 0.333 0.667 1.000 0.000 1.333 -1.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 (RHS) 0 0.04068 0.08053 0.11873 0.15452 0.18717 0.21599 0.24042 Inverse Matrix 1 1.000 0.875 0.766 0.670 0.586 0.513 0.449 0.393 2 0.000 0.875 0.766 0.670 0.586 0.513 0.449 0.393 3 0.000 0.000 0.875 0.766 0.670 0.586 0.513 0.449 4 0.000 0.000 0.000 0.875 0.766 0.670 0.586 0.513 5 0.000 0.000 0.000 0.000 0.875 0.766 0.670 0.586 6 0.000 0.000 0.000 0.000 0.000 0.875 0.766 0.670 7 0.000 0.000 0.000 0.000 0.000 0.000 0.875 0.766 8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.875 Result 0.000 0.036 0.102 0.193 0.304 0.430 0.565 0.705 Exact 0.000 0.099 0.346 0.669 Error 0.000 0.001 0.002 0.001 0.066 N=8 Δx = 0.143 x 0.000 0.143 0.286 0.429 0.571 0.714 0.857 1.000 Exact 0.000 0.019 0.074 0.157 0.264 0.389 0.526 0.669 Error 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.032 N = 16 Δx = 0.067 x 0.000 0.067 0.133 0.200 0.267 0.333 0.400 0.467 0.533 0.600 0.667 0.733 0.800 0.867 0.933 1.000 N 4 8 16 2 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 4 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 6 0.000 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 7 8 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 0.000 0.000 -1.000 1.067 0.000 0.000 -1.000 1.067 0.000 0.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 11 12 13 14 15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 16 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 (RHS) 0 0.00888 0.01773 0.02649 0.03514 0.04363 0.05192 0.05999 0.06779 0.07529 0.08245 0.08925 0.09565 0.10162 0.10715 0.1122 Inverse Matrix 1.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.380 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Eq. 5.34 (LHS) 1 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.380 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 Result 0.000 0.008 0.024 0.048 0.078 0.114 0.155 0.202 0.253 0.308 0.366 0.426 0.489 0.554 0.620 0.686 Δx 0.333 0.143 0.067 Error 0.066 0.032 0.016 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 Exact 0.000 0.004 0.017 0.037 0.065 0.099 0.139 0.184 0.234 0.288 0.346 0.407 0.470 0.535 0.602 0.669 Error 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.016 0.0 0.2 0.4 N=4 N=8 N = 16 x 0.6 Exact solution Actual Error Least Squares Fit .01 0.10 Δx 1.00 0.8 1.0 Problem 5.89 New Eq. 5.34: − ui −1 + (1 + Δx )ui = Δx ⋅ xi2 N=4 Δx = 0.333 Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 1.333 -1.000 0.000 0.000 0.000 1.333 (RHS) 2 0.03704 0.14815 0.33333 Inverse Matrix 1.000 0.750 0.563 0.422 0.000 0.750 0.563 0.422 0.000 0.000 0.750 0.563 0.000 0.000 0.000 0.750 Result 2.000 1.528 1.257 1.193 Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 x 0.000 0.333 0.667 1.000 0.000 1.333 -1.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.143 (RHS) 2 0.00292 0.01166 0.02624 0.04665 0.07289 0.10496 0.14286 Inverse Matrix 1 1.000 0.875 0.766 0.670 0.586 0.513 0.449 0.393 2 0.000 0.875 0.766 0.670 0.586 0.513 0.449 0.393 3 0.000 0.000 0.875 0.766 0.670 0.586 0.513 0.449 4 0.000 0.000 0.000 0.875 0.766 0.670 0.586 0.513 5 0.000 0.000 0.000 0.000 0.875 0.766 0.670 0.586 6 0.000 0.000 0.000 0.000 0.000 0.875 0.766 0.670 7 0.000 0.000 0.000 0.000 0.000 0.000 0.875 0.766 8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.875 Result 2.000 1.753 1.544 1.374 1.243 1.151 1.099 1.087 Exact 2.000 1.444 1.111 1.000 Error 0.000 0.002 0.005 0.009 0.128 N=8 Δx = 0.143 x 0.000 0.143 0.286 0.429 0.571 0.714 0.857 1.000 Exact 2.000 1.735 1.510 1.327 1.184 1.082 1.020 1.000 Error 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.057 N = 16 Δx = 0.067 x 0.000 0.067 0.133 0.200 0.267 0.333 0.400 0.467 0.533 0.600 0.667 0.733 0.800 0.867 0.933 1.000 N 4 8 16 2 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 4 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 6 0.000 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 7 8 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 0.000 0.000 -1.000 1.067 0.000 0.000 -1.000 1.067 0.000 0.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 -1.000 0.000 0.000 0.000 0.000 0.000 11 12 13 14 15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 1.067 0.000 0.000 0.000 0.000 -1.000 16 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.067 (RHS) 2 0.0003 0.00119 0.00267 0.00474 0.00741 0.01067 0.01452 0.01896 0.024 0.02963 0.03585 0.04267 0.05007 0.05807 0.06667 Inverse Matrix 1.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.380 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Eq. 5.34 (LHS) 1 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.380 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.405 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.432 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.461 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.492 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.524 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 Result 2.000 1.875 1.759 1.652 1.553 1.463 1.381 1.309 1.245 1.189 1.143 1.105 1.076 1.056 1.044 1.041 Δx 0.333 0.143 0.067 Error 0.128 0.057 0.027 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.559 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.679 0.637 0.597 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.724 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.772 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 0.824 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.879 Exact 2.000 1.871 1.751 1.640 1.538 1.444 1.360 1.284 1.218 1.160 1.111 1.071 1.040 1.018 1.004 1.000 Error 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.027 0.0 0.2 0.4 x N=4 N=8 N = 16 0.6 Exact solution 0.8 Actual Error Least Squares Fit .01 0.10 Δx 1.00 1.0 Problem 5.90 Equation of motion: M u du du A = − μA = −μ δ dt dy du ⎛ μA ⎞ +⎜ ⎟u = 0 dt ⎝Mδ ⎠ du + k ⋅u = 0 dt New Eq. 5.34: − u i −1 + (1 + k ⋅ Δ x )u i = 0 N=4 Δt = 0.333 A= δ= μ= M= k= Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 2.067 -1.000 0.000 0.000 0.000 2.067 Inverse Matrix 1.000 0.484 0.234 0.113 0.000 0.484 0.234 0.113 0.000 0.000 0.484 0.234 0.000 0.000 0.000 0.484 Result 1.000 0.484 0.234 0.113 Eq. 5.34 (LHS) 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 t 0.000 0.333 0.667 1.000 0.000 2.067 -1.000 0.000 (RHS) 1 0 0 0 0.000 1.457 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.457 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.457 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.457 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.457 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.457 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.457 2 0.000 0.686 0.471 0.323 0.222 0.152 0.104 0.072 3 0.000 0.000 0.686 0.471 0.323 0.222 0.152 0.104 4 0.000 0.000 0.000 0.686 0.471 0.323 0.222 0.152 5 0.000 0.000 0.000 0.000 0.686 0.471 0.323 0.222 6 0.000 0.000 0.000 0.000 0.000 0.686 0.471 0.323 7 0.000 0.000 0.000 0.000 0.000 0.000 0.686 0.471 8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.686 Result 1.000 0.686 0.471 0.323 0.222 0.152 0.104 0.072 m2 mm 2 0.4 N.s/m 5 kg -1 3.2 s (RHS) 1 0 0 0 0 0 0 0 Inverse Matrix 1 1.000 0.686 0.471 0.323 0.222 0.152 0.104 0.072 0.01 0.25 Exact 1.000 0.344 0.118 0.041 Error 0.000 0.005 0.003 0.001 0.098 N=8 Δt = 0.143 t 0.000 0.143 0.286 0.429 0.571 0.714 0.857 1.000 Exact 1.000 0.633 0.401 0.254 0.161 0.102 0.064 0.041 Error 0.000 0.000 0.001 0.001 0.000 0.000 0.000 0.000 0.052 N = 16 Δt = 0.067 t 0.000 0.067 0.133 0.200 0.267 0.333 0.400 0.467 0.533 0.600 0.667 0.733 0.800 0.867 0.933 1.000 N 4 8 16 2 0.000 1.213 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 0.000 0.000 1.213 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 4 0.000 0.000 0.000 1.213 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 5 0.000 0.000 0.000 0.000 1.213 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 6 0.000 0.000 0.000 0.000 0.000 1.213 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 7 8 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.213 0.000 0.000 -1.000 1.213 0.000 0.000 -1.000 1.213 0.000 0.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.213 -1.000 0.000 0.000 0.000 0.000 0.000 11 12 13 14 15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.213 0.000 0.000 0.000 0.000 -1.000 1.213 0.000 0.000 0.000 0.000 -1.000 1.213 0.000 0.000 0.000 0.000 -1.000 1.213 0.000 0.000 0.000 0.000 -1.000 1.213 0.000 0.000 0.000 0.000 -1.000 16 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.213 (RHS) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Inverse Matrix 1.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.119 0.098 0.081 0.067 0.055 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Eq. 5.34 (LHS) 1 1.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.119 0.098 0.081 0.067 0.055 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.119 0.098 0.081 0.067 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.119 0.098 0.081 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.119 0.098 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.119 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 Result 1.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.145 0.119 0.098 0.081 0.067 0.055 Δt 0.333 0.143 0.067 Error 0.098 0.052 0.027 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.175 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.313 0.258 0.213 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.380 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.461 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 0.560 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.824 0.679 Exact 1.000 0.808 0.653 0.527 0.426 0.344 0.278 0.225 0.181 0.147 0.118 0.096 0.077 0.062 0.050 0.041 Error 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.027 1.2 N=4 N=8 N = 16 Exact solution 1.0 u (m/s) 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 t (s) 1 Actual Error Least Squares Fit ε 0.1 0.01 0.01 0.10 Δx 1.00 0.8 1.0 Problem 5.91 ui = Δx = 2 u g i −1 + Δx u g i 1 + 2 Δx u g i 0.333 x Iteration 0 1 2 3 4 5 6 Exact 0.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.333 1.000 0.800 0.791 0.791 0.791 0.791 0.791 0.750 0.667 1.000 0.800 0.661 0.650 0.650 0.650 0.650 0.600 1.000 1.000 0.800 0.661 0.560 0.550 0.550 0.550 0.500 Residuals 0.204 0.127 0.068 0.007 0.000 0.000 1E+00 1.0 1E-01 1E-02 1E-03 Residual R Iterations = 2 Iterations = 4 Iterations = 6 Exact Solution 0.9 1E-04 0.8 1E-05 u 1E-06 0.7 1E-07 1E-08 0.6 1E-09 0.5 1E-10 0 1 2 3 Iteration N 4 5 6 0.0 0.2 0.4 0.6 x 0.8 1.0 Problem 5.92 ui = Δx = 2 ug i −1 + Δx ug i 1 + 2Δx ug i 0.0667 Iteration 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.067 1.000 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.941 0.133 1.000 0.941 0.889 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.888 0.200 1.000 0.941 0.889 0.842 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.841 0.267 1.000 0.941 0.889 0.842 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.799 0.333 1.000 0.941 0.889 0.842 0.799 0.761 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.400 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 0.725 x 0.467 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.693 0.533 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.664 0.600 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.637 0.667 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.637 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.612 0.733 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.637 0.612 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.589 0.800 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.637 0.612 0.589 0.568 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.567 0.867 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.637 0.612 0.589 0.568 0.548 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.547 0.933 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.637 0.612 0.589 0.568 0.548 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 0.529 1.000 1.000 0.941 0.889 0.842 0.799 0.761 0.726 0.694 0.664 0.637 0.612 0.589 0.568 0.548 0.529 0.512 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 0.511 Exact 1.000 0.938 0.882 0.833 0.789 0.750 0.714 0.682 0.652 0.625 0.600 0.577 0.556 0.536 0.517 0.500 1.0 Iterations = 10 Iterations = 20 Iterations = 30 Exact Solution 0.9 0.8 u 0.7 0.6 0.5 0.0 0.2 0.4 0.6 x 0.8 1.0 Problem 5.93 ui − ui −1 1 + =0 ui Δx Δui = ui − u g i 1 1 1 ⎛ Δui ⎞ ⎜1 − ⎟ = ≈ ui ug i + Δui u g i ⎜ ug i ⎟ ⎝ ⎠ ui − ui −1 1 ⎛ ui − ug i ⎜1 − + ug i ⎜ ug i Δx ⎝ ui − ui −1 1 ⎛ ⎜ 2 − ui + Δx ug i ⎜ ug i ⎝ Δx = ⎛ Δx ⎞ 2Δx ui ⎜1 − 2 ⎟ = ui −1 − ⎜ u⎟ ug i gi ⎠ ⎝ 2Δx ui −1 − ug i ui = Δx 1− 2 ug i ⎞ ⎟=0 ⎟ ⎠ ⎞ ⎟=0 ⎟ ⎠ 0.667 x Iteration 0 1 2 3 4 5 6 Exact Δx = 0.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 0.667 2.000 1.600 1.577 1.577 1.577 1.577 1.577 1.633 1.333 2.000 1.600 1.037 0.767 1.211 0.873 0.401 1.155 2.000 2.000 1.600 1.037 -0.658 -5.158 1.507 -0.017 0.000 0.133 Iteration 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 0.133 2.000 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 1.931 0.267 2.000 1.931 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 1.859 0.400 2.000 1.931 1.859 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 1.785 0.533 2.000 1.931 1.859 1.785 1.707 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 1.706 0.667 2.000 1.931 1.859 1.785 1.707 1.625 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 1.624 0.800 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 1.538 x 0.933 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.445 1.067 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.346 1.200 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.242 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.239 1.333 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.242 1.124 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.120 1.467 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.242 1.124 0.991 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 0.984 1.600 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.242 1.124 0.991 0.836 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 0.822 1.733 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.242 1.124 0.991 0.836 0.639 0.601 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 0.599 1.867 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.242 1.124 0.991 0.836 0.639 0.329 0.899 0.363 9.602 0.572 0.225 0.359 3.969 0.537 0.191 0.300 0.600 0.246 0.403 -0.345 -11.373 0.623 0.261 0.442 -0.013 -0.027 -0.059 -0.136 -0.414 5.624 0.554 0.209 0.329 0.919 0.367 -11.148 0.624 0.262 0.443 -0.010 -0.019 -0.041 -0.090 -0.231 -1.171 0.916 0.366 -18.029 0.614 0.256 0.426 -0.097 2.000 2.000 1.931 1.859 1.785 1.707 1.625 1.539 1.447 1.348 1.242 1.124 0.991 0.836 0.639 0.329 2.061 0.795 0.034 -0.016 -0.034 -0.070 -0.160 -1.332 0.797 -0.182 -0.584 1.734 0.097 0.178 0.572 -19.981 0.637 -0.234 -1.108 0.255 1.023 -0.366 132.420 -0.416 27.391 0.545 -0.510 1.749 0.802 0.044 0.252 0.394 -2.929 0.542 -0.918 0.322 3.048 -0.180 -0.402 -2.886 1.025 0.122 2.526 0.520 -0.509 1.962 Exact 2.000 1.932 1.862 1.789 1.713 1.633 1.549 1.461 1.366 1.265 1.155 1.033 0.894 0.730 0.516 0.000 2.5 Iterations = 2 Iterations = 4 Iterations = 6 Exact Solution 2.0 1.5 u 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 x 2.5 Iterations = 20 Iterations = 40 Iterations = 60 Exact Solution 2.0 1.5 u 1.0 0.5 0.0 0.0 0.5 1.0 x 1.5 2.0 Problem 5.94 du 2 = k (U − u ) dt v =U −u dv = − du M dv = kv 2 dt dv k 2 v =0 + dt M −M Δt = k= M= 1.000 10 70 2 vi2 ≈ 2v g i vi − v g i vi − vi −1 k 2 + 2vg i vi − vg i = 0 Δt M k 2 Δt v g i v g i −1 + M vi = k 1 + 2 Δt v g i M ( ) N.s2/m2 kg t Iteration 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 0 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 7.500 1 7.500 4.943 4.556 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 4.547 2 7.500 4.943 3.496 3.153 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3.139 3 7.500 4.943 3.496 2.623 2.364 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 2.350 4 7.500 4.943 3.496 2.623 2.061 1.870 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 1.857 5 7.500 4.943 3.496 2.623 2.061 1.679 1.536 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 1.525 6 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.297 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 1.288 7 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.119 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 1.112 8 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.982 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.976 9 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.930 0.874 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 0.868 10 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.930 0.832 0.786 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 0.781 11 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.930 0.832 0.752 0.713 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 0.709 12 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.930 0.832 0.752 0.686 0.653 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 0.649 13 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.930 0.832 0.752 0.686 0.629 0.601 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 0.598 14 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.930 0.832 0.752 0.686 0.629 0.581 0.557 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 0.554 15 7.500 4.943 3.496 2.623 2.061 1.679 1.407 1.205 1.051 0.930 0.832 0.752 0.686 0.629 0.581 0.540 0.519 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 0.516 Above values are for v! To get u we compute u = U - v Iteration 10 20 40 0.000 0.000 0.000 2.953 2.953 2.953 4.361 4.361 4.361 5.150 5.150 5.150 5.643 5.643 5.643 5.975 5.975 5.975 6.212 6.212 6.212 6.388 6.388 6.388 6.524 6.524 6.524 6.626 6.632 6.632 6.668 6.719 6.719 6.668 6.791 6.791 6.668 6.851 6.851 6.668 6.902 6.902 6.668 6.946 6.946 6.668 6.984 6.984 Exact 0.000 3.879 5.114 5.720 6.081 6.320 6.490 6.618 6.716 6.795 6.860 6.913 6.959 6.998 7.031 7.061 8 7 u (m/s) 6 5 Iterations = 10 Iterations = 20 Iterations = 40 Exact Solution 4 3 2 1 0 0 2 4 6 8 t (s) 10 12 14 16 ...
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