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Transonic flows are complex problems. At cer tain Mach nu mbers, in gas flows t here are local transitions from the subsonic into the superson ic and reciprocally, especially the discontinuity throughout shock waves. Subsonic flows and supersonic flows have very different properties, thus calcu lat ion methods are also different the subsonic equation is elliptic and the supersonic equation is hyperbolic. Using established programs (based on the incompressibl e theory, the compressible theory and the transonic theory), we studi ed sub-t ransonic flow characteristics with variations of the Mach number. Our studies are a lso carried out with variations of incidence, whi ch impact on t he transonic effect . Numerical results show that the local compressibility depends on not only the speed but also the incidence. At a rather great angle of attack, local supersonic regions can appear with free Mach numbers in the order of 0.3 t he Mach number is normal ly considered as a speed of incompressible flows. According to these results, we can recognize for wh ich regime the local compressibili ty of flow is considerable and then it is necessary to consider the problem under t he transonic flow t heory. 1. INTROD UCTION The transonic flow is a mixed flow in which there are locally subsonic and supersonic regions, normally wit h free stream Mach numbers from 0.7 to 1.2 (Fig. 1). However , experiences and calculations show that transonic flow properties depend also on the body geometry. By using established programs to calculate flows around a profile, it depend on incidences, a supersonic point can occur even Mach number NI ::; 0.3. The transonic flow is a complex problem, especially when taking into account t he viscosity, the problern needs always-experimental results. \ i\Tith advantages of sub-transonic flows (JV! < 1), they are generally used in the civil transportation. In the world, they study transonic flows experimentally and numerically. In our country, studies of transonic and supersonic flows are a beginning. vVe realized and publicized some computational study works for supersonic flows, and t hen transonic flows . In t his work, we approach numerically t he ste2.1y isentropic potential plan problem and compare the calculated results with published results. Numerical essays show more clear the nature of t he transonic flow and influences of fluid compressibili ty on aerodynamic characteristics under the transonic t heory. Since t hen, we can see optimal and efficacy application domains of the flow type. 498 Hoang Thi Eich Ngoc .,.. .. ,,, ,/ '.,,. ~ Ji /' l ~ ~ ·> t/ ""'O < l/ M ~~ l i'"'1"' I. M > 1 I \ \M>! \ \ ,, ... Fig. 1. Transonic flows free Mach number M 0 < 1 and M 0 > 1 2. ESTABLISHED PROGRAMS We programmed sub-transonic problems by three ways: 1. based on incompressible fl.ow theory; 2. based on compressible flow theory; 3. based on transonic fl.ow theory. 2 .1. Methods for incompressible and compressible flow calculations A code for incompressible flows is programmed according to singularity method with linear vortexes [1] . At a point P, formulas calculating velocities induced from a vortex distribution of strength I = 10+11.x can be written as fo llowings ( 10 and 11 are constant) [2]: IO ( 1 Y 1 Y ) up = tan tan 2n X X2 X X1 [ ( ) 2 2 '( ) ] 11 X X1 + Y . 1 Y 1 Y + z . ln ( )2 2 + 2x tan tan , 47f X X2 + y X X2 X X1 10 1 (x x1) 2 + y2 w = n --P 4n (x x2)2+y2 11 [x (x x1) 2 +y 2 ( 1 Y 1 Y ) ] ln +(x1 x2)+y tan tan 2n 2 (x x2) 2 + y72 x x2 x x1 (2 .1) vVe divide the profile contour into N panels with law x = ~ ( 1 cos a) to narrow grids near the leading edge and t he trailing edge, correspondingly there are N conditions on the surface and a Joukovski condition. Linear equations are written under following matrix:

Influences of the compressibility on aerodynamic characteristics of profile under the transonic flow theory