Determination of critical distributed roughness height inducing transition on AEDC cone at transonic/supersonic speed in CIRA PT-1 Wind Tunnel

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1. TRANSITION ON AEDC CONE 
In this chapter the aims of the experiment performed in CIRA PT-1 transonic wind 
tunnel will be introduced. In particular, a briefly explanation of laminar to turbulent 
transition theory will be reported in order to present the subject of the current 
experiment. A literary investigation of transition phenomenon on AEDC cone will be 
drafted and foundations for MDOE technique application will be drawn up. Finally 
practical application of experimental results in CIRA PT-1 transonic wind tunnel will 
be mentioned  
1.1 Laminar to turbulent transition 
In order to understand the transition from laminar to turbulent flow it is necessary 
to introduce the concept of boundary-layer. Considering a real fluid motion along a thin 
flat plate, the influence of viscosity at high Reynolds numbers is confined to a very thin 
layer in the immediate neighbourhood of the solid wall. This effect is displayed by 
fluid adherence to the wall caused by frictional forces which retard the motion of the 
fluid in a thin layer near the wall. In this small region flow velocity increases from zero 
at the wall (no slip condition) to its full value equal to which one of external 
frictionless flow. The layer mentioned is called boundary layer. 
The thickness of boundary layer increases along the plate from the leading edge in 
downstream direction because quantity of fluid affected by motion delay increases. 
Evidently the thickness of boundary layer reduces with decreasing viscosity. The next 
figure (Figure 1) represents velocity distribution in such a boundary layer on flat-plate 
with dimensions considerably exaggerated.
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Figure 1: Sketch of velocity distribution in boundary layer on a flat plate in parallel 
flow at zero incidence 
 
In front of the leading edge of the plate velocity distribution is uniform. When 
boundary layer increases, even if small viscosity is present, the frictional shearing 
stresses y u ∂ ∂ = μ τ are considerable because of the large velocity gradient across the 
flow. Outside the boundary layer these frictional shearing stresses are very small, so the 
flow field can be imaginary subdivided into two regions: the thin boundary layer near 
the wall, in which friction must be taken into account, and the region outside the 
boundary layer, where the forces due to frictional shearing stress are small and may be 
neglected. 
Within the boundary layer it can occur an amplification of certain small 
disturbances assumed already present in laminar flow. These disturbances can be 
originated by wall roughness or due to irregularities in the external flow. If 
disturbances decay with time, the main flow is considered stable; whereas if 
disturbances increase with time, the flow is considered unstable and there exists the 
possibility of transition to a turbulent pattern. 
Transition effect happens when fluid particles ceases to move along straight lines 
and the regularity of the motion breaks down; fluctuations are superimposed on the
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basic flow motion and disturbance amplification causes fluid particles mix. This non-
linear instationary phenomenon is known as laminar to turbulent transition and it takes 
place in a region at definite value of Reynolds number (called critical) depending on 
details of the experimental arrangements, in particular on the amount of disturbance 
suffered by the flow before its run over the body. Today the main theory used to 
explain the laminar-turbulent transition is Tollmien-Schlichting: progressive waves 
originate in critical layer and develop with time making flow unstable. According to 
the theory of stability, this is due to the viscosity effect which make flow unstable at 
critical Reynolds number in a particular region called critical layer. Since disturbances 
are present in all the free flow, but are amplified only in this critical layer, the cause of 
the magnification is considered lying in a variation of kinetic energy density expressed 
by Reynolds-Orr equation (Ref.[1]). 
In general, transition in boundary layer on a solid body in a stream is affected by 
many parameters such as pressure distribution in external flow, nature of the wall 
(roughness) and disturbances in free flow (intensity of turbulence).  
In turbulent flow conditions, velocity distribution near the wall is even much grater 
than in laminar flow because of the intense momentum exchange inside the boundary 
layer, near the wall. Furthermore, turbulent mixing dissipates a large quantity of energy 
which causes considerably increase of resistance against flow. Moreover it has been 
demonstrated that rough surface causes considerably higher values of skin friction in 
turbulent flow compared with smooth ones. The same effect is true for heat transfer 
coefficient, but the percentage increase in the rate of heat transfer is smaller than that in 
skin friction. This difference is easily understood since a part of turbulent shearing 
stresses can be transmitted to the wall trough pressure forces exerted on protuberances, 
but an analogue mechanism does not exist for heat transfer.
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 Transition on AEDC cone from literature 
The AEDC cone geometry is probably the most used 3D configuration for laminar-
turbulent transition experiment in wind tunnel. Then many literature data are available 
from specific experiments with this geometry.  
Taking into account the aims of the current activity, it is essential to understand the 
effect of several parameters such as Mach number, transition tripping position, 
concentrated roughness height and Reynolds number on transition position on AEDC 
cone. 
Furthermore, in literature the transition position on the AEDC cone is represented 
by Xt/L (with respect to cone length) or by the variable ReX
T
 which links the transition 
position to Reynolds number. In this manner it is possible to have information on 
transition phenomenon in a more general way, regardless of specific cone geometry 
order, and to extend results to generic 3D geometry. 
The results of this literary research are reported below. 
1.2.1 Mach number effect on transition position  
Mach number is certainly an important parameter for transition position 
determination. Figure 2 (Ref.[2]) shows the Reynolds number estimated at the end of 
transition region as a function of Mach number, on a 10°-cone in the NTF (National 
Transonic Facility at NASA Langley). In the figure R
T
 corresponds to ReX
T
 before 
defined.
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Figure 2: Estimated end of transition Reynolds number as a function of the Mach 
number on a 10°-cone in the NTF. 
 
Figure 2 shows that the ReX
T
 at which transition occurs rises with Mach. In 
particular, it is possible to notice an initial constant trend followed by a rapid increase 
of transition Reynolds number. A critical Mach number (M=0.6) can be identified, 
where this trend change happens. 
1.2.2 Trip Transition height effect on transition position 
Trip Transition height is certainly another important parameter on transition 
position. Despite it can be expected that the trip transition size influence the transition,
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this effect is not so widely described in literature on AEDC cone and a well known 
mathematical law relating roughness height and transition position is not available. 
There are very few transition measurement results pertaining to roughness elements 
locally distributed on a surface, so it is very difficult to find a reference to establish a 
relation between trip transition height and transitional Reynolds number. 
In Ref.[3] is reported that when the Reynolds number calculated with the roughness 
height k
s
 exceeds 
120
1
=
v
k U
s
, the critical Reynolds number dramatically drops and this 
value determines the critical roughness size. 
 
Figure 3: Ratio of the critical Reynolds number at a flat plate at zero incidence with 
single roughness elements to that of the smooth plate, after H.I. Dryden (1953)
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Figure 3 (Ref.[3]), shows the ratio of critical Reynolds number on a flat plate at 
zero incidence with single roughness elements to Reynolds number of the smooth plate, 
respect to trip transition height (referred to the boundary layer thickness). In 
particularly, when the trip transition height increases, the Reynolds number ratio 
decreases.  
For the current activity, it is also useful to consider the correlation between 
minimum roughness size inducing transition and critical roughness Reynolds number, 
formulated with the local flow conditions about the particles, reported in Ref.[8]. In 
particular roughness particles smaller than the critical value induce no disturbance of 
sufficient magnitude, in boundary layer, to induce transition. Whereas roughness 
particles equal to the critical size initiate the formation of turbulent spots near the 
roughness that coalesce into a continuously turbulent flow somewhat downstream of 
the roughness. In this condition , only a small increase in roughness Reynolds number 
above the critical value is required to move the fully developed turbulent boundary 
layer substantially up to the roughness particles. Once defined Reynolds number based 
on the length of x from leading edge to roughness station, and the condition outside the 
boundary layer, for a given Mach number, unit Reynolds number and roughness 
location, the nondimensional roughness height may be determined as showed in Figure 
4. Lets notice that currently this method is widely used at PT-1 wind tunnel to chose 
the critical roughness height inducing transition to be install on models.