Ch.4 Stationary Normal Shockwaves

2. Ch.4 Stationary Normal Shockwaves

3. 4.1 Introduction 1.Definition of Shock Wave and Normal Shock Wave] 1] Shock Wave * The shock process represents an abrupt change in fluid properties in which finite variations in pressure, temperature, and density occur over a shock thickness comparable to the mean free path of the gas molecules involved. * The extremely thin region in which the transition from the supersonic velocity, relatively low pressure state to the state that involves a relatively low velocity and high pressure is termed a shock wave. * finite pressure disturbance, compressive elastic wave of finite strength * nonlinear addition of the individual Mach waves emitted from each point on the body of finite thickness (or finite attack angle, not point projectile). * The shock wave is an abrupt disturbance that causes discontinuous and irreversible changes in such fluid properties as speed, which changes from supersonic to lower speeds, pressure, temperature, and density. shock waves

4. 2] Normal Shock Wave * a shock wave which is straight with the flow at right angles to the wave. * a plane surface of discontinuity normal to flow direction * stationary normal shock waves and moving normal speed, which changes from supersonic to lower speeds, pressure, temperature, and density.([18] p138) * This case represents the simplest example of a shock in that changes in flow properties occur only in the direction of flow; thus it can be treated with the equations of one-dimensional gas dynamics.

6. 4.2 Formation of a Normal Shockwave

7. 1. Case of Formation 1) Internal Flow When a supersonic flow decelerates in response to a sharp increase in pressure. (e.g. ; flow in de Laval nozzle ) 2) External Flow When a supersonic flow encounters a sudden compressive change in flow direction. (e.g. ; concave corner )

13. Normal shock wave at M=1.5 A pattern of pairs of weak oblique shock waves is produced by strips of tape on the floor and ceiling of a supersonic nozzel. They terminate at an almost straight and normal shock wave, showing that the flow is subsonic downstream. U.S. Air Force photograph, courtesy of Arnold Engineering Development Center

15. 미해군 곡예비행단의 F/A-18. 쇼의 일환으로 마하 0.85 가량으로 비행중. 주변 영역에 초음속 흐름이 생김에 따라 구름이 생성됨.

17. 날개 단면을 컴퓨터로 해석한 그림. 마하 0.8 정도로 비행중인 날개를 묘사. 주황색 부분은 마하 1 이상인 부분이며, 빨간색으로 갈 수록 속도가 빨라짐. 옆의 표에서 확인할 수 있듯 일정 부분은 최대 마하 1.44까지 올라감. 빨간색 공기흐름이 갑자기 급격히 사라지는 부분이 바로 수직으로 충격파가 생긴 부분.

18. Cone-cylinder in supersonic free flight A cone-cylinder of 12.5 deg. semi-vertex angle is shot through air at M=1.84. The boundary layer becomes turbulent shortly behind the vertex, and generates Mach waves that are visible in this shadowgraph.Photographs by A.C. Charters

19. Sphere at M=4.01 This shadowgraph of a 1/2-inch sphere in free flight through atmosphereic air shows boundary layer separation just behind the equator, accompanied by a weak shock wave, and formation of the N-wave that is heard as a double boom far away. The vertical line is a reference cord. Photograph by A.C. Charters

20. Model in wind tunnel

21. Fired pistol. Schlieren photograph of the muzzle blast of a .22 calibre pistol: propellent gas in wake of bullet and spherical shock wave.

22. Shuttle in wind tunnel. Shuttle airflow. Schlieren photograph of an early design of the space shuttle undergoing a transonic wind tunnel test.

23. 2. Characteristics 1] irreversible, adiabatic process As a result of the gradients in temperature and velocity that are create by a shock, heat is transferred and energy is dissipated within the gas, and these processes are the thermodynamically irreversible. ([18] p138) 2] discontinuities in properties 3] thickness ; [ ]; order of several molecular mean free paths 4] shape ; A shock wave is, in general, curved. However, many shock

24. waves that occur in practical situations are straight, being either at right angles to or at an angle to the upstream flow. • 5] summary([18] p156) • - remains constant and is equal to 1. • decreases and approaches zero as approaches infinity. • increase with increasing • 3. Difference between a shock wave and a sound wave • 1] sound wave • * infinitesimal deflection of the stream due to the body • (=infinitesimal pressure disturbance) • *A Mach wave represents a surface across which some derivatives of the flow variables may be discontinuous while the

25. variables themselves are continuous. So the characteristic curves (or Mach lines) are patching lines for continuous flow. • * Governing eq. ⇒ differential form • 2] shock wave • * finite deflection of the stream due to the body = finite pressure disturbance = nonlinear addition of the sound wave • * A shock wave represents a surface across which the thermodynamic properties and the flow velocity as well as their derivatives are essentially discontinuous. So shock waves are patching lines for discontinuous flow • * across shock waves. • * Governing eq. ⇒ integral form • * Shock waves propagate faster than Mach waves do, and they

26. show large gradients in pressure and in density.([18] p140) 4. Formation of Shock (see [T] p81 Fig.5.8)

29. 5. Expansion Wave (팽창파, 膨脹波)

31. 4.3 Equations governing a stationary normal shockwave 1. Basic Assumptions * one-dimensional steady flow (=stationary wave) * frictionless flow (=no boundary layer) * Shock process takes place at constant area. * shock wave ⊥ streamlines * adiabatic flow without external work, negligible body force 2. Governing Equations 1] control volume Fig. 4.7 2] unknown variables

32. 3] Jump Conditions

33. 4. M1 – M2 Relation from Continuity Equation

35. 5. Proof of Irreversible Process 1]

36. 2] Fig. 4.10 So expansion shock is impossible.

38. 6. Normal Shock Wave Relations in terms of Mach Number • * Eq. of state, speed of sound • * Adiabatic process = isoenergetic

40. ; Since will be between 1 and 2, and since is always greater than 1, it follows from this equation that will always be less than 1, i.e., the flow downstream of a normal shock wave will always be subsonic. * (see [1] p71 Fig. 4.10, [18] p148, Fig. 4.7)

41. 7. Rankine-Hugoniot Normal Shock Wave Relation 1] The set of equations which give in terms of the shock strength is often termed the Rankine-Hugoniot shock wave relations.

42. 2] Rankine-Hugoniot Curve for • At low pressure ratios, the Hugoniot curve and isentropic curve differ only slightly from each other, so that a weak shock appears like an isentropic process. - At the large pressure ratios which characterize strong shocks, the density ratio reaches the limiting value of ; in isentropic process, however, the density ratio increases constantly.

43. 3] Remarks - Shock wave must always be compressive, i.e. , that must be greater than 1, i.e., the pressure must always increase across the shock wave. And the density always increases, the velocity always decreases, and the temperature always increases across a shock wave.

44. 8. Other Important Normal Shock Relation for a Calorically Perfect Gas in terms of and 1] Properties ratio between in front of and behind the shock wave * ; often termed the strength of the shock wave

46. 9. Normal Shock Table

47. M2<1

48. Example 4.1 . Limiting Cases of Normal Shock Wave Relations 1] Shock Strength 2] Very Strong Normal Shock ; a normal shock wave for which ; Thus, if tends to infinity, and tends to infinity but xxxxx and tend to finite values. So the assumption that the gas remains thermally and calorically perfect will cease to be valid when the shock is very strong because very high temperatures will then usually exist behind the shock.

49. 3] Very Weak Normal Shock Wave ( ; ) • The weak shock relations apply if . • 1)

50. Shock Wave