Uncertainty in airflow field parameters in a study of shock waves on flat plate in transonic wind tunnel

When flat plates are submitted to wind tunnel tests and the upstream Mach number is greater than 1.0 this gives rise to shock and expansion waves. This study experimentally evaluates the airflow field parameters ahead of and behind the shock region and compares the data obtained in the tests with the expected theoretical results. For the case of expansion waves only theoretical values are supplied in this preliminary part of the study. The drag and lift forces acting on the model being tested are also estimated from the pressure values measured on the plate surface. Data reduction includes evaluation of the uncertainties associated with the measured quantities, as well as their propagation to the output quantities, employing the Monte Carlo method. Numerical simulation of the phenomena using Computational Fluid Dynamics is also done. Data obtained analytically, experimentally and numerically are compared.


Introduction
Shock waves occur in nature when an object moves faster than the speed of sound and when the presence of the object causes an abrupt decrease in the airflow area. If the airflow area increases, a different phenomenon is observed: an expansion fan [1,2]. Shock and expansion waves cause changes in the gas properties and this is the aim of the present study: to experimentally evaluate the conditions associated with the shock and expansion caused by the presence of the flat plate in the airflow. Aerodynamic forces acting on the plate are also calculated. The uncertainties in the estimated flow field parameters are obtained using the Monte Carlo method. The mean value and standard deviation of measured parameters are computed and this information is used to determine the probability density function for output quantities [3]. The study of flat plates is important because they have typical characteristics of all supersonic airfoils. The inclined flat plate is the simplest case of lift airfoil. The tests were carried out in the TTP pilot transonic wind tunnel facility. In the first part of the test campaign, the flat plate was positioned at an angle of attack equal to -4 o . The angle between the direction of the incident airflow and the direction of the plate is called angle of attack α.

The TTP pilot transonic wind tunnel
The wind tunnel TTP is located at the Aerodynamics Division of the Institute of Aeronautics and Space, IAE/ALA/TTP (figure 1a). The tunnel is continuously driven by an 830 kW main axial compressor and can also be operated intermittently by means of an injection system, which supplies airflow for around 30 seconds. It is a variable-pressure wind tunnel with control capability to independently vary Mach number, stagnation pressure, stagnation temperature and humidity. The test section is 0.25 m high and 0.30 m wide and has longitudinally slotted walls to favor the uniformity of the airflow. At TTP, the Mach number range can vary from 0.20 to 1.30. Some configurations can be changed in order to allow better flow control such as reentry flaps position and the rate of forced mass extraction, determined by the Plenum Evacuation system, PES. A schematic picture of the aerodynamic circuit is presented in figure 1b, highlighting its main parts: the axial compressor, cooler, stilling chamber, plenum chamber, high velocity diffuser and corners (numbered from 1 to 4). Arrows indicate the direction of the airflow. The test section, which receives the test article, is located inside the plenum chamber.
a) The circuit.
b) The schematic diagram.

Methodology
In order to evaluate the airflow parameters ahead of the shock and expansion waves caused by the presence of the plate, the freestream airflow static pressure, p 1 , and total pressure, p 0 , were measured. The value of p 1 was obtained at the first pressure tap of the wind tunnel superior wall (figure 5a) and the measurement of p 0 was taken in the plenum chamber (figure 5b). The mean value and standard deviation of the pressure signals, p 0 and p 1 , supplied by the instruments during the runs of the wind tunnel tests are computed. This information is used to determine the probability density function for Mach number using the Monte Carlo method [3].  The static pressure behind the shock and expansion wave regions is estimated through the average value plate p calculated from the six static pressure taps located on the flat plate surface.
The angle of the leading edge of flat plate is equal to +8 o . When positioning the plate at an angle of attack α equal to -4 o , the airflow encounters the superior surface of the plate at +4 o .
In the following sections we will separate the aerodynamic phenomena which take place on the superior and inferior surfaces of the flat plate.
The uncertainties in the estimated parameters are evaluated by the Monte Carlo method. MatLab® codes were elaborated for data reduction. The number of trials employed is 500,000. The distribution assigned to the input quantities is the Gaussian.

Superior surface
In the superior surface of the plate, the airflow encounters a concave corner and the streamlines are deflected upward through the angle θ equal to +4º. An oblique shock occurs in this case (figure 6).
The angle between the shock wave and the upstream flow direction is defined as the wave angle and is denoted by β.
Knowing the flow field parameters in region 1 ahead of the shock, one can estimate the parameters in region 2, behind the shock.
The equation relating the deflection angle, θ, the Mach number, M, in region 1 and the shock wave angle, β, is [1]: where γ is equal to 1.4 for air at normal conditions.
The normal component of the of the upstream Mach number, M n,1 , is: The pressure ratio across the shock wave is: ( ) The total pressure in region 2, p 0,2 , is evaluated using the equation:

Inferior surface
On the inferior surface of the plate, the airflow encounters a convex corner and an expansion wave is formed. Such wave is denoted as Prandtl-Meyer expansion wave (figure 7). Expansion is an isentropic phenomenon and therefore the total pressure is constant across the expansion. The numerical code was based on finite difference and structured mesh with first-order implicit time discretization and second-order spatial central discretization. The code follows the main characteristics of the Alternating Direction Implicit scheme, ADI, and linear artificial viscosity. The mesh is composed by two juxtaposed blocks. Each block has 191 points in the longitudinal direction and 51 points in the vertical direction and the Δy minimum is equal to 0.002 m. Inlet boundary conditions were imposed, outlet boundary conditions were extrapolated by zero-th order, slip condition were imposed on wall surfaces together with adiabatic wall consideration and pressure extrapolation. The test section slotted walls were simulated alternating far field static pressure and solid wall configurations. The force coefficients, C F , are calculated using the equation:

Forces in the flat plate
In equation (13), F represents the lift force, L, or the drag force, D, S is the reference area and q ∞ is the dynamic pressure of the freestream flow, given by:

Superior surface
The mean value of the static pressures, measured by the tap pressures located on the flat plate, is   Equation 10 supplies the static pressure p 3 = 41905.5 Pa. The uncertainty value is u p3 = 583.4 Pa. As the tests for measuring p 3 have not yet been carried out, there is no comparison between experimental and theoretical values in this paper.

Inferior surface
The angle between the forward Mach line and the upstream airflow direction is μ 1 = 56.0 o and the angle between the rearward Mach line and the flat plate direction is μ 2 = 47.8º. The angle between the rearward Mach line and the upstream airflow direction is μ 2 -θ = 43.8º (figure 7).
Numerical results for Mach number distribution of the flow field using CFD were also obtained and were presented in figure 8.  [5], which presents the value 0.17 for the rate C D /C L 2 . The value obtained in this study is 0.15.

Conclusions
This paper presents the results obtained in the tests with the flat plate in the transonic wind tunnel TTP, in the supersonic regime. In this range of airflow speed, shock and expansion waves are expected. The parameters static pressure, p 1 , and total pressure, p 0 , of the freestream flow field were measured. From the measurement of these parameters, it was possible to evaluate the changes in the airflow properties due to the presence of the plate. The six pressure taps on the superior flat plate surface supply the experimental data for the comparison between the measured static pressure and that predicted by the shock wave theory. The drag and lift forces acting on the plate were also estimated experimentally. The uncertainties in the estimated flow field parameters were estimated through the Monte Carlo method.
In both cases, static pressure and aerodynamic forces, the experimental results comply with literature.
Measurements of parameters such as temperature of the freestream airflow and total pressure behind the shock, as well as static pressure behind the expansion, will be performed in future tests.