SPATER ONE DIMENSIONAL COMPRESSIBLE FLOW Continuity Equation for steady flow
SPATER ONE DIMENSIONAL COMPRESSIBLE FLOW EULER EQUATION For steady flow
SPATER ONE DIMENSIONAL COMPRESSIBLE FLOW Momentum Equation Energy Equation
SPATER Oblique Shock Waves The oblique shock waves typically occurs when a supersonic flow is turned to itself by a wall or its equivalent boundary condition. All the streamlines have the same deflection angle at the shock wave, parallel to the surface downstream. Across the oblique shock, M decreases but p, T and increase.
SPATER Expansion Waves The expansion waves typically occur when a supersonic flow is turned away from itself by a wall or its equivalent boundary condition. The streamlines are smoothly curved through the expansion fan until they are all parallel to the wall behind the corner point. All flow properties through an expansion wave change smoothly and continuously. Across the expansion wave, M increases while p, T, and decreases.
SPATER Source of Oblique Waves For an object moving at a supersonic speed, the object is always ahead of the sound wave fronts generated by the object. This cause the sound wave fronts to coalesce into a line disturbance, called Mach wave, at the Mack angle relative to the direction of the beeper. The physical mechanism to form the oblique shock wave is essentially the same as the Mach wave. The Mach wave is actually an infinitely weak shock wave.
SPATER Oblique Shock Relations The oblique shock tilts at a wave angle with respect to V 1, the upstream velocity. Behind the shock, the flow is deflected toward the shock by the flow deflection angle Let u and w denote the normal and parallel flow velocity components relative to the oblique shock and M n and M t the corresponding Mach numbers, we have for a steady adiabatic flow with no body forces the following relations:
SPATER Oblique Shock Relations ( contd.) So and M n1 and M n2 all satisfy the corresponding normal shock relations, which are all functions of M 1 and, because
SPATER Oblique Shock Relations ( contd.) θ-β-M relation 1.For any given free stream Mach number M 1, there is a maximum beyond which the shock will be detached. 2.For any given M 1 and < max, there are two s. The larger is called the strong shock solution, where M 2 is subsonic. The lower is called the weak shock solution, where M 2 is supersonic except for a small region near max. 3. If =0, then = /2 (normal shock) or = (Mach wave).
SPATER Straight Oblique Shock Relations ( contd.) For a calorically perfect gas,
SPATER Supersonic Flow Over Cones The flow over a cone is inherently three- dimensional. The three-dimensionality has the relieving effect to result in a weaker shock wave as compared to a wedge of the same half angle. The flow between the shock and the cone is no longer uniform; the streamlines there are curved and the surface pressure are not constant.
SPATER Shock Wave Reflection Consider an incident oblique shock on a lower wall that is reflected by the upper wall at point. The reflection angle of the shock at the upper wall is determined by two conditions: (a) M 2 < M 1 (b) The flow downstream of the reflected shock wave must be parallel to the upper wall. That is, the flow is deflected downward by.
SPATER Pressure-Deflection Diagram The pressure-deflection diagram is a plot of the static pressure behind an oblique shock versus the flow deflection angle for a given upstream condition. For left-running waves, the flow deflection angle is upward; it is considered as positive. For right- running waves, the flow deflection angle is downward; it is considered as negative.
SPATER Intersection of Shock Waves of Opposite Families Consider the intersection of left- and right-running shocks (A and B). The two shocks intersect at E and result in two refracted shocks C and D. Since the shock wave strengths of A and B in general are different, there is a slip line in the region between the two refracted waves where (a) the pressure is continuous but the entropy is discontinuous at the slip line; (b) the velocities on two sides of the slip line are in the same direction but of different magnitudes;
SPATER Intersection of Shock Waves of the Same Family As two left running oblique shock waves A and B intersect at C, they will form a single shock wave CD and a reflected shock wave CE such that there is slip line in the region between CD and CE.
SPATER Prandtl-Meyer Expansion Waves M 2 > M 1. An expansion corner is a means to increase the flow mach number. P 2 /p 1 <1, 2 / 1 <1, T 2 /T 1 < 1. The pressure, density, and temperature decrease through an expansion wave. The expansion fan is a continuous expansion region, composed of of an infinite number of Mach waves, bounded upstream by 1 and downstream by 2.
SPATER Prandtl-Meyer Expansion Waves ( contd.) Centered expansion fan is also called Prandtl- Meyer expansion wave. where 1 = sin -1 (1/M 1 ) and 2 = sin -1 (1/M 2 ). Streamlines through an expansion wave are smooth curved lines. Since the expansion takes place through a continuous succession of Mach waves, and ds = 0 for each wave, the expansion is isentropic.
SPATER Prandtl-Meyer Expansion Waves ( contd.) For perfect gas, the Prandtl-Meyer expansion waves are governed by Knowing M 1 and 2, we can find M
SPATER Prandtl-Meyer Expansion Waves ( contd.) Since the expansion is isentropic, and hence To and Po are constant, we have
SPATER Shock-Expansion Method -Flow Conditions Downstream of the Trailing Edge In supersonic flow, the conditions at the trailing edge cannot affect the flow upstream. Therefore, unlike the subsonic flow, there is no need to impose a Kutta condition at the trailing edge in order to determine the airfoil lift. However, if there is an interest to know the flow conditions downstream of the T.E., they can be determined by requiring the pressures downstream of the top- and bottom-surface flows to be equal.
SPATER Conditions Downstream of the T.E. -An Example For the case shown, the angle of attack is less than the wedges half angle so we expect two oblique shocks at the trailing edge. In order to know the flow conditions downstream of the airfoil, we start a guess value of the deflection angle of the downstream flow relative of the free stream. Knowing the Mach number and static pressure immediately upstream of each shock leads to the prediction of the static pressures downstream of each shock. Then through the iteration process, g is changed until the pressures downstream of the top- and bottom-surface flow become equal.
SPATER Total and Perturbation Velocity Potentials Consider a slender body immersed in an inviscid, irrotational flow where We can define the (total) velocity potential and the perturbation velocity potential as follows:
SPATER Velocity Potential Equation For a steady, irrotational flow, starting from the differential continuity equation we have In terms of the velocity potential (x,y,z), the above continuity equation becomes
SPATER Linearized Velocity Potential Equation By assuming small velocity perturbations such that we can prove that for the Mach number ranges excluding the transonic range: the hypersonic range:
SPATER Linearized Pressure Coefficient For calorically perfect gas, the pressure coefficient C p can be reduce to For small velocity perturbations, we can prove that Note that the linearized C p only depends on u.
SPATER Prandtl-Glauert Rule for Linearized Subsonic Flow (2-D Over Thin Airfoils)
SPATER C p of 2-D Supersonic Flows Around Thin Wings For supersonic flow over any 2-D slender airfoil, where is the local surface inclination with respect to the free stream:
SPATER C l of 2-D Supersonic Flow Over Thin Wings For supersonic flow over any 2-D slender airfoil,
SPATER C m of 2-D Supersonic Flow Over Thin Wings For supersonic flow over any 2-D slender airfoil, the pitching moment coefficient with respect to an arbitrary point x o is The center of pressure for a symmetrical airfoil in supersonic flow is predicted at the mid-chord point.