Phy 352: Fluid Dynamics, Spring 2013

Transcription

1 Phy 352:, Spring 2013 Prasad Indian Institute of Science Education and Research (IISER), Pune

2 Module 5 Trans-sonic flows in astrophysics

3 Module 5 Trans-sonic flows in astrophysics The solar wind, spherical accretion onto compact objects

4 Module 5 Trans-sonic flows in astrophysics The solar wind, spherical accretion onto compact objects Jets from compact objects: the de Laval nozzle

5 Module 5 Trans-sonic flows in astrophysics The solar wind, spherical accretion onto compact objects Jets from compact objects: the de Laval nozzle Supernova shocks - the Sedov-Taylor self-similar solution

6 Module 5 Trans-sonic flows in astrophysics The solar wind, spherical accretion onto compact objects Jets from compact objects: the de Laval nozzle Supernova shocks - the Sedov-Taylor self-similar solution

7 Trans-sonic flows Simply put, flows that transition from being sub-sonic to supersonic

8 Trans-sonic flows Simply put, flows that transition from being sub-sonic to supersonic...or supersonic to subsonic

9 Trans-sonic flows Simply put, flows that transition from being sub-sonic to supersonic...or supersonic to subsonic...generally exhibit counter-intuitive features

10 Trans-sonic flows Simply put, flows that transition from being sub-sonic to supersonic...or supersonic to subsonic...generally exhibit counter-intuitive features We first consider outflows from/accretion onto a spherical object

11 Spherical geometry Mass conservation: ρur 2 = Constant; equivalently,

12 Spherical geometry Mass conservation: ρur 2 = Constant; equivalently, 1dρ ρ dr + 1 u du dr + 2 r = 0

13 Spherical geometry Mass conservation: ρur 2 = Constant; equivalently, 1dρ ρ dr + 1 u du dr + 2 r = 0 Momentum equation (inviscid flow, but gravity is important!)

14 Spherical geometry Mass conservation: ρur 2 = Constant; equivalently, 1dρ ρ dr + 1 u du dr + 2 r = 0 Momentum equation (inviscid flow, but gravity is important!) ρu du dr + dp dr = ρgm r 2

15 Spherical geometry Mass conservation: ρur 2 = Constant; equivalently, 1dρ ρ dr + 1 u du dr + 2 r = 0 Momentum equation (inviscid flow, but gravity is important!) ρu du dr + dp dr = ρgm r 2 Use the sound speed to eliminate p: dp/dρ = c 2 s

16 Spherical geometry Mass conservation: ρur 2 = Constant; equivalently, 1dρ ρ dr + 1 u du dr + 2 r = 0 Momentum equation (inviscid flow, but gravity is important!) ρu du dr + dp dr = ρgm r 2 Use the sound speed to eliminate p: dp/dρ = c 2 s Combine eqs to get (work it out!) ( ) u c2 s du u dr = 2c2 s r GM r 2

17 Trans-sonic solutions

18 Trans-sonic solutions

19 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point;

20 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point; i.e., u = c s, then

21 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point; i.e., u = c s, then 2c 2 s r = GM r 2

22 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point; i.e., u = c s, then 2c 2 s r = GM r 2 why? The location of the sonic point is given by

23 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point; i.e., u = c s, then 2c 2 s r = GM r 2 why? The location of the sonic point is given by 2c 2 s r s = GM r 2 s

24 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point; i.e., u = c s, then 2c 2 s r = GM r 2 why? The location of the sonic point is given by 2c 2 s r s = GM r 2 s What would be du/dr at the sonic point?

25 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point; i.e., u = c s, then 2c 2 s r = GM r 2 why? The location of the sonic point is given by 2c 2 s r s = GM r 2 s What would be du/dr at the sonic point? In general, one needs to shoot appropriately

26 Different kinds of trans-sonic solutions If the flow needs to pass through a sonic point; i.e., u = c s, then 2c 2 s r = GM r 2 why? The location of the sonic point is given by 2c 2 s r s = GM r 2 s What would be du/dr at the sonic point? In general, one needs to shoot appropriately (i.e., right combination of u and du/dr) at the base in order to get it through the sonic point

27 Trans-sonic solutions

28 Trans-sonic solutions Winds; e.g., the solar wind, and spherically symmetric accretion flows

29 The solar wind Predicted by Eugene Parker (1958)

30 The solar wind Predicted by Eugene Parker (1958) The pressure at the base of the million degree corona is so high that it drives a transonic solar wind

31 The solar wind Predicted by Eugene Parker (1958) The pressure at the base of the million degree corona is so high that it drives a transonic solar wind Among the few instances where the theoretical prediction predated the observation!

32 The solar wind Predicted by Eugene Parker (1958) The pressure at the base of the million degree corona is so high that it drives a transonic solar wind Among the few instances where the theoretical prediction predated the observation! Parker s seminal paper was rejected by the referee!

33 The solar wind Predicted by Eugene Parker (1958) The pressure at the base of the million degree corona is so high that it drives a transonic solar wind Among the few instances where the theoretical prediction predated the observation! Parker s seminal paper was rejected by the referee! We are all immersed in the solar wind, so to speak; it continues on until the heliopause

34 Blowin in the wind: comet tails

35 Blowin in the wind: comet tails

36 The Parker spiral, heliospheric current sheet

37 The Parker spiral, heliospheric current sheet

38 Heliopause, termination shock

39 Heliopause, termination shock

40 Onward to 1D flows Flows through a 1D channel

41 Onward to 1D flows Flows through a 1D channel ; e.g., flow through a pipe, astrophysical jets, etc

42 Onward to 1D flows Flows through a 1D channel ; e.g., flow through a pipe, astrophysical jets, etc We already know something about this:

43 Onward to 1D flows Flows through a 1D channel ; e.g., flow through a pipe, astrophysical jets, etc We already know something about this: Mass conservation: ρau = constant

44 Onward to 1D flows Flows through a 1D channel ; e.g., flow through a pipe, astrophysical jets, etc We already know something about this: Mass conservation: ρau = constant...and the Bernoulli constant

45 Onward to 1D flows Flows through a 1D channel ; e.g., flow through a pipe, astrophysical jets, etc We already know something about this: Mass conservation: ρau = constant...and the Bernoulli constant c 2 s + γ 1 2 u 2 = constant

46 Onward to 1D flows Flows through a 1D channel ; e.g., flow through a pipe, astrophysical jets, etc We already know something about this: Mass conservation: ρau = constant...and the Bernoulli constant c 2 s + γ 1 2 u 2 = constant how does the flow behave in a diverging/converging channel?

47 Lets do it a little differently.. Differential form of mass continuity equation:

48 Lets do it a little differently.. Differential form of mass continuity equation: 1 dρ ρ dx + 1 u du dx + 1 da A dx = 0

49 Lets do it a little differently.. Differential form of mass continuity equation: 1 dρ ρ dx + 1 u du dx + 1 da A dx = 0...and the differential form of the (inviscid) momentum equation: ρu du dx + dp dx = 0

50 Lets do it a little differently.. Differential form of mass continuity equation: 1 dρ ρ dx + 1 u du dx + 1 da A dx = 0...and the differential form of the (inviscid) momentum equation: ρu du dx + dp dx = 0 Use c 2 s = dp/dρ and combine these two eqs:

51 Lets do it a little differently.. Differential form of mass continuity equation: 1 dρ ρ dx + 1 u du dx + 1 da A dx = 0...and the differential form of the (inviscid) momentum equation: ρu du dx + dp dx = 0 Use c 2 s = dp/dρ and combine these two eqs: (show!)

52 Lets do it a little differently.. Differential form of mass continuity equation: 1 dρ ρ dx + 1 u du dx + 1 da A dx = 0...and the differential form of the (inviscid) momentum equation: ρu du dx + dp dx = 0 Use c 2 s = dp/dρ and combine these two eqs: (show!) (M 2 1) 1 u du dx = 1 da A dx

53 Lets do it a little differently.. Differential form of mass continuity equation: 1 dρ ρ dx + 1 u du dx + 1 da A dx = 0...and the differential form of the (inviscid) momentum equation: ρu du dx + dp dx = 0 Use c 2 s = dp/dρ and combine these two eqs: (show!) (M 2 1) 1 u du dx = 1 da A dx There s a lot hidden in here!

54 Converging/diverging channels (M 2 1) 1 u du dx = 1 da A dx

55 Converging/diverging channels Consider M < 1. (M 2 1) 1 u du dx = 1 da A dx

56 Converging/diverging channels (M 2 1) 1 u du dx = 1 da A dx Consider M < 1. u, A > 0, so when when da/dx < 0 (converging channel), does the flow accelerate (du/dx > 0) or decelerate (du/dx < 0)?

57 Converging/diverging channels (M 2 1) 1 u du dx = 1 da A dx Consider M < 1. u, A > 0, so when when da/dx < 0 (converging channel), does the flow accelerate (du/dx > 0) or decelerate (du/dx < 0)? So subsonic flows do conform to intuition, but what about supersonic flows (M > 1)?

58 Converging/diverging channels (M 2 1) 1 u du dx = 1 da A dx Consider M < 1. u, A > 0, so when when da/dx < 0 (converging channel), does the flow accelerate (du/dx > 0) or decelerate (du/dx < 0)? So subsonic flows do conform to intuition, but what about supersonic flows (M > 1)? For M > 1, the signs of da/dx and du/dx are the same!

59 Converging/diverging channels (M 2 1) 1 u du dx = 1 da A dx Consider M < 1. u, A > 0, so when when da/dx < 0 (converging channel), does the flow accelerate (du/dx > 0) or decelerate (du/dx < 0)? So subsonic flows do conform to intuition, but what about supersonic flows (M > 1)? For M > 1, the signs of da/dx and du/dx are the same! So a supersonic flow decelerates in a converging channel and accelerates in a diverging channel!

60 Nozzles/diffusers

61 Nozzles/diffusers

62 The de Laval nozzle

63 The de Laval nozzle Transonic flow; sonic point where da/dx = 0

64 Flow need not always be transonic

65 Flow need not always be transonic Other extrema also possible, also shocks

66 Astrophysical jets - I

67 Astrophysical jets - I

68 Astrophysical jets - II

69 Astrophysical jets - II

70 Spherical blast waves - I We ll investigate energy deposition

71 Spherical blast waves - I We ll investigate energy deposition (a large amount of energy)

72 Spherical blast waves - I We ll investigate energy deposition (a large amount of energy) at a point

73 Spherical blast waves - I We ll investigate energy deposition (a large amount of energy) at a point in a uniform, unbounded medium

74 Spherical blast waves - I We ll investigate energy deposition (a large amount of energy) at a point in a uniform, unbounded medium e.g., a bomb explosion

75 Spherical blast waves - I We ll investigate energy deposition (a large amount of energy) at a point in a uniform, unbounded medium e.g., a bomb explosion a supernova explosion

76 Spherical blast waves - I We ll investigate energy deposition (a large amount of energy) at a point in a uniform, unbounded medium e.g., a bomb explosion a supernova explosion This problem was investigated independently by Sedov (1946, 1959) in the Soviet Union and Taylor (1950) in the US in connection with nuclear bomb explosions

77 Spherical blast waves - I We ll investigate energy deposition (a large amount of energy) at a point in a uniform, unbounded medium e.g., a bomb explosion a supernova explosion This problem was investigated independently by Sedov (1946, 1959) in the Soviet Union and Taylor (1950) in the US in connection with nuclear bomb explosions Much of this research was classified, and it was openly published (at least in the west) only after the end of World war II

78 Spherical blast waves - II Amount of energy deposited is so large that it produces an outward propagating, spherical shock

79 Spherical blast waves - II Amount of energy deposited is so large that it produces an outward propagating, spherical shock Assume that the shock propagates outward in a self-similar fashion;

80 Spherical blast waves - II Amount of energy deposited is so large that it produces an outward propagating, spherical shock Assume that the shock propagates outward in a self-similar fashion; i.e., the (spatial) configuration at a later time appears like an enlarged/shrunken version of the configuration at an initial time

81 Spherical blast waves - II Amount of energy deposited is so large that it produces an outward propagating, spherical shock Assume that the shock propagates outward in a self-similar fashion; i.e., the (spatial) configuration at a later time appears like an enlarged/shrunken version of the configuration at an initial time Self-similarity is usually a good assumption for the energy conserving/adiabatic phase of the blast wave evolution

82 Spherical blast waves - II Amount of energy deposited is so large that it produces an outward propagating, spherical shock Assume that the shock propagates outward in a self-similar fashion; i.e., the (spatial) configuration at a later time appears like an enlarged/shrunken version of the configuration at an initial time Self-similarity is usually a good assumption for the energy conserving/adiabatic phase of the blast wave evolution We study supernova remnants as an illustrative example

83 Supernovae Stellar explosions:

84 Supernovae Stellar explosions: 1 Core of an aging massive star undergoes gravitational collapse, releasing a burst of gravitational potential energy

85 Supernovae Stellar explosions: 1 Core of an aging massive star undergoes gravitational collapse, releasing a burst of gravitational potential energy 2 A white dwarf can accrete enough material from a companion to ignite runaway carbon fusion at its core

86 Supernovae Stellar explosions: 1 Core of an aging massive star undergoes gravitational collapse, releasing a burst of gravitational potential energy 2 A white dwarf can accrete enough material from a companion to ignite runaway carbon fusion at its core Supernova explosions

87 Supernovae Stellar explosions: 1 Core of an aging massive star undergoes gravitational collapse, releasing a burst of gravitational potential energy 2 A white dwarf can accrete enough material from a companion to ignite runaway carbon fusion at its core Supernova explosions 1 Enrich the interstellar medium with heav(ier) elements

88 Supernovae Stellar explosions: 1 Core of an aging massive star undergoes gravitational collapse, releasing a burst of gravitational potential energy 2 A white dwarf can accrete enough material from a companion to ignite runaway carbon fusion at its core Supernova explosions 1 Enrich the interstellar medium with heav(ier) elements (all of us have been inside a star)

89 Supernovae Stellar explosions: 1 Core of an aging massive star undergoes gravitational collapse, releasing a burst of gravitational potential energy 2 A white dwarf can accrete enough material from a companion to ignite runaway carbon fusion at its core Supernova explosions 1 Enrich the interstellar medium with heav(ier) elements (all of us have been inside a star) 2 Shock fronts can accelerate cosmic rays

90 Supernovae Stellar explosions: 1 Core of an aging massive star undergoes gravitational collapse, releasing a burst of gravitational potential energy 2 A white dwarf can accrete enough material from a companion to ignite runaway carbon fusion at its core Supernova explosions 1 Enrich the interstellar medium with heav(ier) elements (all of us have been inside a star) 2 Shock fronts can accelerate cosmic rays 3 Expanding shock front can trigger new star formation

91 Tycho s supernova remnant - multiwavelength

92 Tycho s supernova remnant - radio

93 The Crab nebula

94 CasA

95 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1

96 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation

97 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation ; i.e., E remains constant w/ time

98 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation ; i.e., E remains constant w/ time Ram pressure of shock front ρ 1 U 2 sh ambient pressure p 1;

99 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation ; i.e., E remains constant w/ time Ram pressure of shock front ρ 1 U 2 sh ambient pressure p 1; i.e., neglect p 1

100 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation ; i.e., E remains constant w/ time Ram pressure of shock front ρ 1 U 2 sh ambient pressure p 1; i.e., neglect p 1 The only two relevant quantities, then: E and ρ 1

101 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation ; i.e., E remains constant w/ time Ram pressure of shock front ρ 1 U 2 sh ambient pressure p 1; i.e., neglect p 1 The only two relevant quantities, then: E and ρ 1 Let λ be a quantity (with the dimensions of length) which gives the scale of the blast after time t:

102 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation ; i.e., E remains constant w/ time Ram pressure of shock front ρ 1 U 2 sh ambient pressure p 1; i.e., neglect p 1 The only two relevant quantities, then: E and ρ 1 Let λ be a quantity (with the dimensions of length) which gives the scale of the blast after time t: The only way to form a quantity with the dimensions of length from E, ρ 1 and t is:

103 Main assumptions A (large amount of) energy E is injected at a point into a medium of density ρ 1 Neglect any energy losses due to radiation ; i.e., E remains constant w/ time Ram pressure of shock front ρ 1 U 2 sh ambient pressure p 1; i.e., neglect p 1 The only two relevant quantities, then: E and ρ 1 Let λ be a quantity (with the dimensions of length) which gives the scale of the blast after time t: The only way to form a quantity with the dimensions of length from E, ρ 1 and t is: ( Et 2 λ = ρ 1 ) 1/5

104 The self-similarity variable ξ Consider the (dimensional) radius of the blast wave r(t) at time t

105 The self-similarity variable ξ Consider the (dimensional) radius of the blast wave r(t) at time t This can be expressed in units of λ as

106 The self-similarity variable ξ Consider the (dimensional) radius of the blast wave r(t) at time t This can be expressed in units of λ as ξ = r ( ) 1/5 λ = r ρ1 Et 2

107 The self-similarity variable ξ Consider the (dimensional) radius of the blast wave r(t) at time t This can be expressed in units of λ as ξ = r ( ) 1/5 λ = r ρ1 Et 2 The dimensionless similarity variable ξ:

108 The self-similarity variable ξ Consider the (dimensional) radius of the blast wave r(t) at time t This can be expressed in units of λ as ξ = r ( ) 1/5 λ = r ρ1 Et 2 The dimensionless similarity variable ξ: 1 labels each radial shell; i.e., it doesn t change for a given shell

109 The self-similarity variable ξ Consider the (dimensional) radius of the blast wave r(t) at time t This can be expressed in units of λ as ξ = r ( ) 1/5 λ = r ρ1 Et 2 The dimensionless similarity variable ξ: 1 labels each radial shell; i.e., it doesn t change for a given shell 2 is a unique combination of r and t

110 The self-similarity variable ξ Consider the (dimensional) radius of the blast wave r(t) at time t This can be expressed in units of λ as ξ = r ( ) 1/5 λ = r ρ1 Et 2 The dimensionless similarity variable ξ: 1 labels each radial shell; i.e., it doesn t change for a given shell 2 is a unique combination of r and t (can you think of a similar combination of r and t you have seen earlier?)

111 How does the shock front expand? Using the similarity parameter ξ

112 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t)

113 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time?

114 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple;

115 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple; let ξ shock label the shock front; i.e.,

116 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple; let ξ shock label the shock front; i.e., r shock (t) = ξ shock ( Et 2 ρ 1 ) 1/5

117 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple; let ξ shock label the shock front; i.e., r shock (t) = ξ shock ( Et 2 ρ 1 ) 1/5..so this predicts that the shock front spreads out as t 2/5 ;

118 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple; let ξ shock label the shock front; i.e., r shock (t) = ξ shock ( Et 2 ρ 1 ) 1/5..so this predicts that the shock front spreads out as t 2/5 ; is it borne out by observations?

119 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple; let ξ shock label the shock front; i.e., r shock (t) = ξ shock ( Et 2 ρ 1 ) 1/5..so this predicts that the shock front spreads out as t 2/5 ; is it borne out by observations? Also, the velocity of shock expansion is

120 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple; let ξ shock label the shock front; i.e., r shock (t) = ξ shock ( Et 2 ρ 1 ) 1/5..so this predicts that the shock front spreads out as t 2/5 ; is it borne out by observations? Also, the velocity of shock expansion is v shock = dr shock dt =

121 How does the shock front expand? Using the similarity parameter ξ (which contains both r and t) can we figure out how the shock front will expand with time? Simple; let ξ shock label the shock front; i.e., r shock (t) = ξ shock ( Et 2 ρ 1 ) 1/5..so this predicts that the shock front spreads out as t 2/5 ; is it borne out by observations? Also, the velocity of shock expansion is v shock = dr shock dt = 2 5 ξ 0 ( ) E 1/5 ρ 1 t 3

122 Blast wavefront Data from nuclear explosion in New Mexico, 1945

123 Structure of the blast wave Having figured the time evolution of the shock front (r shock t 2/5 ), we now ask what s behind the front

124 Structure of the blast wave Having figured the time evolution of the shock front (r shock t 2/5 ), we now ask what s behind the front..how does the shocked gas look like?

125 Structure of the blast wave Having figured the time evolution of the shock front (r shock t 2/5 ), we now ask what s behind the front..how does the shocked gas look like?..we have the usual jump conditions for a strong shock: (1 undisturbed medium, 2 behind the shock/ shocked medium) ρ 2 = γ +1 ρ 1 γ 1, u 2 = 2 u shock γ +1,

126 Structure of the blast wave Having figured the time evolution of the shock front (r shock t 2/5 ), we now ask what s behind the front..how does the shocked gas look like?..we have the usual jump conditions for a strong shock: (1 undisturbed medium, 2 behind the shock/ shocked medium) ρ 2 = γ +1 ρ 1 γ 1, u 2 = 2 u shock γ +1, (show)

127 Structure of the blast wave Having figured the time evolution of the shock front (r shock t 2/5 ), we now ask what s behind the front..how does the shocked gas look like?..we have the usual jump conditions for a strong shock: (1 undisturbed medium, 2 behind the shock/ shocked medium) and ρ 2 = γ +1 ρ 1 γ 1, u 2 = 2 u shock γ +1, (show)

128 Structure of the blast wave Having figured the time evolution of the shock front (r shock t 2/5 ), we now ask what s behind the front..how does the shocked gas look like?..we have the usual jump conditions for a strong shock: (1 undisturbed medium, 2 behind the shock/ shocked medium) ρ 2 = γ +1 ρ 1 γ 1, u 2 = 2 u shock γ +1, (show) and p 2 = 2 γ +1 ρ 1u 2 shock

129 Structure of the blast wave Having figured the time evolution of the shock front (r shock t 2/5 ), we now ask what s behind the front..how does the shocked gas look like?..we have the usual jump conditions for a strong shock: (1 undisturbed medium, 2 behind the shock/ shocked medium) ρ 2 = γ +1 ρ 1 γ 1, u 2 = 2 u shock γ +1, (show) and p 2 = 2 γ +1 ρ 1u 2 shock (show)

130 non-dimensional variables.. ρ(r,t) = ρ 2 ρ (ξ)

131 non-dimensional variables.. ρ(r,t) = ρ 2 ρ (ξ) = ρ 1 γ +1 γ 1 ρ (ξ)

132 non-dimensional variables.. ρ(r,t) = ρ 2 ρ (ξ) = ρ 1 γ +1 γ 1 ρ (ξ) u(r,t) = u 2 r r shock u (ξ)

133 non-dimensional variables.. ρ(r,t) = ρ 2 ρ (ξ) = ρ 1 γ +1 γ 1 ρ (ξ) u(r,t) = u 2 r r shock u (ξ) = 4 r 5(γ +1) t u (ξ)

134 non-dimensional variables.. ρ(r,t) = ρ 2 ρ (ξ) = ρ 1 γ +1 γ 1 ρ (ξ) u(r,t) = u 2 r r shock u (ξ) = ( ) r 2 p(r,t) = p 2 p (ξ) r shock 4 r 5(γ +1) t u (ξ)

135 non-dimensional variables.. ρ(r,t) = ρ 2 ρ (ξ) = ρ 1 γ +1 γ 1 ρ (ξ) u(r,t) = u 2 r r shock u (ξ) = ( ) r 2 p(r,t) = p 2 p (ξ) = r shock 4 r 5(γ +1) t u (ξ) 8ρ 1 25(γ +1) ( ) r 2 p (ξ) t

136 non-dimensional variables.. ρ(r,t) = ρ 2 ρ (ξ) = ρ 1 γ +1 γ 1 ρ (ξ) u(r,t) = u 2 r r shock u (ξ) = ( ) r 2 p(r,t) = p 2 p (ξ) = r shock 4 r 5(γ +1) t u (ξ) 8ρ 1 25(γ +1) ( ) r 2 p (ξ) t The dimensionless variables will be used in the usual (mass, momentum and energy) conservation equations:

137 (Dimensional) conservations Eqs in spherical geometry Mass conservation:

138 (Dimensional) conservations Eqs in spherical geometry Mass conservation: ρ t + 1 r 2 r (r2 ρu) = 0,

139 (Dimensional) conservations Eqs in spherical geometry Mass conservation: ρ t + 1 r 2 r (r2 ρu) = 0, Momentum conservation (inviscid flows)

140 (Dimensional) conservations Eqs in spherical geometry Mass conservation: ρ t + 1 r 2 r (r2 ρu) = 0, Momentum conservation (inviscid flows) u t u +u r = 1 p ρ r,

141 (Dimensional) conservations Eqs in spherical geometry Mass conservation: ρ t + 1 r 2 r (r2 ρu) = 0, Momentum conservation (inviscid flows) u t u +u r = 1 p ρ r,..and energy conservation (familiar with this form?)

142 (Dimensional) conservations Eqs in spherical geometry Mass conservation: ρ t + 1 r 2 r (r2 ρu) = 0, Momentum conservation (inviscid flows) u t u +u r = 1 p ρ r,..and energy conservation (familiar with this form?) ( t +u ) log p r ρ γ = 0

143 Self-similar form Using the definition of the similarity parameter ξ: ( ) 1/5 ρ1 ξ = r Et 2

144 Self-similar form Using the definition of the similarity parameter ξ: ( ) 1/5 ρ1 ξ = r Et 2 we get t = 2 ξ 5 t d dξ

145 Self-similar form Using the definition of the similarity parameter ξ: ( ) 1/5 ρ1 ξ = r Et 2 we get and t = 2 ξ 5 t r = ξ r d dξ d dξ

146 Self-similar form Using the definition of the similarity parameter ξ: we get and ( ) 1/5 ρ1 ξ = r Et 2 t = 2 ξ 5 t r = ξ r d dξ d dξ...so now we have derivatives only in ξ (not in r and t), but it gets better..

147 Non-dimensional conservation equations In terms of the non-d variables and ξ-derivatives, the conservation equations are: ξ dρ dξ + 2 ( 3ρ u +ξ d ) γ +1 dξ (ρ u ) = 0 u 2 5 ξ du dξ + 4 5(γ +1) ξ d (log p dξ ) (u 2 +u ξ du 2(γ 1) = dξ 5ρ (γ +1) ) 5(γ +1) 4u = ρ γ 2u (γ +1) Coupled ordinary differential equations in ξ ) (2p +ξ dp dξ

148 Non-dimensional conservation equations In terms of the non-d variables and ξ-derivatives, the conservation equations are: ξ dρ dξ + 2 ( 3ρ u +ξ d ) γ +1 dξ (ρ u ) = 0 u 2 5 ξ du dξ + 4 5(γ +1) ξ d (log p dξ ) (u 2 +u ξ du 2(γ 1) = dξ 5ρ (γ +1) ) 5(γ +1) 4u = ρ γ 2u (γ +1) ) (2p +ξ dp dξ Coupled ordinary differential equations in ξ (instead of the coupled PDEs we started out with)

149 Non-dimensional conservation equations In terms of the non-d variables and ξ-derivatives, the conservation equations are: ξ dρ dξ + 2 ( 3ρ u +ξ d ) γ +1 dξ (ρ u ) = 0 u 2 5 ξ du dξ + 4 5(γ +1) ξ d (log p dξ ) (u 2 +u ξ du 2(γ 1) = dξ 5ρ (γ +1) ) 5(γ +1) 4u = ρ γ 2u (γ +1) ) (2p +ξ dp dξ Coupled ordinary differential equations in ξ (instead of the coupled PDEs we started out with) Boundary conditions ρ (ξ 0 ) = u (ξ 0 ) = p (ξ 0 ) = 0;

150 Non-dimensional conservation equations In terms of the non-d variables and ξ-derivatives, the conservation equations are: ξ dρ dξ + 2 ( 3ρ u +ξ d ) γ +1 dξ (ρ u ) = 0 u 2 5 ξ du dξ + 4 5(γ +1) ξ d (log p dξ ) (u 2 +u ξ du 2(γ 1) = dξ 5ρ (γ +1) ) 5(γ +1) 4u = ρ γ 2u (γ +1) ) (2p +ξ dp dξ Coupled ordinary differential equations in ξ (instead of the coupled PDEs we started out with) Boundary conditions ρ (ξ 0 ) = u (ξ 0 ) = p (ξ 0 ) = 0; ξ 0 shock location

151 One solution Physical conditions in the shocked region for γ = 1.4 (appropriate for air)