CN103969022A - Indirect measuring method for hypersonic speed wind tunnel turbulence scale - Google Patents

Abstract

The invention relates to wind tunnel turbulence scale measuring, in particular to an indirect measuring method for hypersonic speed wind tunnel turbulence scale. According to the indirection measuring method for hypersonic speed wind tunnel turbulence scale, a miniature airspeed head is used for measuring pressure pulsation in a hypersonic speed wind tunnel flow field, and then the turbulence scale is indirectly obtained according to relation conversion between pressure pulsation and speed pulsation. The method includes the steps of wind tunnel data collecting and data analyzing. According to wind tunnel data collection process, an adjustable device of the miniature airspeed tube is used for measuring pulsation pressure values at different positions of wind tunnel incoming flows, and other probes are used for measuring other wind tunnel data. According to the data analyzing process, the measured pressure pulsation values and other wind tunnel data are analyzed to obtain the function relation expression between pressure pulsation and speed pulsation in hypersonic speed airflows through deduction, and therefore the hypersonic speed wind tunnel turbulence scale is obtained through calculation. The function relation expression between pressure pulsation and speed pulsation is simple and clear, the hypersonic speed wind tunnel turbulence scale can be obtained only by pressure pulsation measurement and simple calculation, and the indirect measuring method is convenient and rapid to implement.

Description

A kind of hypersonic wind tunnel turbulivity indirect measurement method

Technical field

The present invention relates to Wind Tunnel Turbulence Spectral Analyzer mobility fields of measurement, particularly relate to a kind of hypersonic wind tunnel turbulivity indirect measurement method.

Background technology

Research discovery, turbulivity is to boundary layer transition important, and it seriously can be compared with Reynolds number the impact of pneumatic gauging.Simultaneously, when utilizing CFD stream field to carry out numerical simulation, turbulivity also plays a part very crucial, and people often need to measure turbulivity or estimate before emulation, and the accuracy of turbulence measurement value or estimated value has directly affected the accuracy of simulation result.

At present, hot-wire anemometer (HWA), Laser Doppler Velocimeter (LDV) and particle image speed-measuring system (PIV) are generally used in the measurement of turbulivity, and they are all by measuring the velocity fluctuation in flow field and then calculating turbulivity.The ultimate principle of HWA is heat balance principle, the fine wire with heating current is placed in flow field, the variation of wind speed can make temperature wiry change, thereby by measure the voltage at hot line two ends can calculate flow velocity (E.L.DOUGHMAN.Development of a Hot-Wire Anemometer for Hypersonic Turbulent Flows[J] .THE REVIEWOF SCIENTIFIC INSTRUMENTS.VOLUME43, NUMBER8,1972).The ultimate principle of LDV is that the Doppler signal of the trace particle by laser probe is measured, and then according to the relation of speed and Doppler frequency, converts and obtains speed.PIV technology is based upon on the basis of flow field display technique, its ultimate principle is in flow field, to add certain trace particle, the test section in the illuminated with laser light flow field of sending with pulsed laser, and with CCD camera, take the image of fluidized particle between test section, image is carried out obtaining velocity distribution (the Pentti Saarenrinne in flow field after analyzing and processing, Mika Piirto and HannuEloranta.Experiences of turbulence measurement with PIV[J] .MEASUREMENT SCIENCE ANDTECHNOLOGY, 2001).

At present, yet there are no the relevant report about method that can Measurement accuracy hypersonic wind tunnel turbulivity both at home and abroad.Above three kinds of measuring methods are in subsonic speed, transonic speed or in low supersonic flow field have a wide range of applications, but they and be not suitable for the turbulence measurement in Hypersonic Flow Field.It is high that HWA has dynamic response frequency, time and spatial resolution advantages of higher, but hot line easily ruptures, and in hypersonic flow field, the environment of high temperature and high speed is an acid test for it.LDV and PIV belong to noncontact FLOW VISUALIZATION technology, and their measuring accuracy are high, and the scope that tests the speed is wide, but in High Speed Flow Field, exist trace particle followability poor, and excited wave affects large shortcoming.

Summary of the invention

Object of the present invention is intended to overcome the existing above-mentioned defect of prior art, provide and utilize miniature pitot to measure the pressure fluctuation in hypersonic wind tunnel flow field, then convert and indirectly obtain a kind of hypersonic wind tunnel turbulivity indirect measurement method of turbulivity according to the relation of pressure fluctuation and velocity fluctuation.

The present invention includes following steps:

One, gather wind tunnel data, concrete grammar is as follows:

1) control motor and make pitot aim at wind-tunnel export center, record the average stagnation pressure of the 1st measuring point with fluctuation pressure p' ₀;

2), by the horizontal or vertical Moving Unit length of pitot, record the pressure data of the 2nd measuring point;

3) continuous repeating step 2), record third and fourth ... the data of individual measuring point, until record all measuring point pressure datas that are evenly distributed on wind-tunnel outlet;

4) near wind-tunnel outlet, along wall, be circumferentially evenly arranged 4 differential pressure pickup and 4 thermopairs of measuring stagnation temperatures of measuring static pressure, record 4 static pressures and 4 stagnation temperature values are averaged and can obtain average static pressure and medial temperature ;

Two, the analysis to wind tunnel data, concrete grammar is as follows:

1) build-up pressure pulsation p' ₀and the funtcional relationship between velocity fluctuation u', density are ρ ';

Step 1 in step 2) in, described build-up pressure pulsation p' ₀and the concrete grammar of the funtcional relationship between velocity fluctuation u', density are ρ ' can be:

In known wind-tunnel, the stagnation pressure expression formula of air-flow is:

$p_0=p+\frac{1}{2}{C}_p{\mathrm{\ρu}}^2---\left(1\right)$

P wherein ₀be stagnation pressure, p is static pressure, represent dynamic pressure, C _pbe pressure coefficient, ρ is the density of air, and u is the speed of air; Pressure coefficient C _pbe the amount relevant with Mach number M to air specific heat ratio γ, its expression formula is:

$C_p=\frac{4}{\mathrm{\γ}+1}(1-\frac{1}{M^2})---\left(2\right)$

By argument table be shown mean value and percent ripple and, even

$p_0=\stackrel{\‾}{p_0}+{p^{\′}}_0;p=\stackrel{\‾}{p}+p^{\′};M=\stackrel{\‾}{M}+M^{\′};\mathrm{\ρ}=\stackrel{\‾}{\mathrm{\ρ}}+{\mathrm{\ρ}}^{\′};u=U+u^{\′}$

Wherein be average static pressure, p' is pulsation static pressure, be average Mach number, M' is Mach number pulsation, be average density, U is average velocity, brings the expression formula of M into formula (2), has

$C_p=\frac{4}{\mathrm{\γ}+1}(1-\frac{1}{{(\stackrel{\‾}{M}+M^{\′})}^2})$

Due under hypersonic m' is very little, and under this functional relation, M' is to C _pimpact very little, therefore ignored, thus pressure coefficient C _pcan be expressed as

$C_p=\frac{4}{\mathrm{\γ}+1}(1-\frac{1}{{\stackrel{\‾}{M}}^2})$

In above formula, air specific heat ratio γ generally gets 1.40, average Mach number there is following relational expression:

$\frac{\stackrel{\‾}{p}}{\stackrel{\‾}{p_0}}={(1+\frac{\mathrm{\γ}-1}{2}{\stackrel{\‾}{M}}^2)}^{-\frac{\mathrm{\γ}}{\mathrm{\γ}-1}}$

Average static pressure in above formula average stagnation pressure therefore be known with air specific heat ratio γ, can calculate thereby can be derived from pressure coefficient C _pvalue;

By each argument table in formula (1) be shown mean value and percent ripple and, formula (1) can be expressed as

$\stackrel{\‾}{p_0}+{p^{\′}}_0=\stackrel{\‾}{p}+p^{\′}+\frac{1}{2}{C}_p(\stackrel{\‾}{\mathrm{\ρ}}{U}^2+\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′2}+2\stackrel{\‾}{\mathrm{\ρ}}U{u}^{\′}+{\mathrm{\ρ}}^{\′}{u}^{\′2}+2{\mathrm{\ρ}}^{\′}U{u}^{\′})$

Due to

$\stackrel{\‾}{p_0}=\stackrel{\‾}{p}+\frac{1}{2}{C}_p\stackrel{\‾}{\mathrm{\ρ}}{\stackrel{\‾}{u}}^2$

Therefore fluctuation pressure can be expressed as:

${p^{\′}}_0=p^{\′}+\frac{1}{2}{C}_p(\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′2}+2\stackrel{\‾}{\mathrm{\ρ}}U{u}^{\′}+{\mathrm{\ρ}}^{\′}{U}^2+{\mathrm{\ρ}}^{\′}{u}^{\′2}+2{\mathrm{\ρ}}^{\′}U{u}^{\′})$

Consider that single order percent ripple is very little, second order percent ripple almost can be ignored, thereby above-mentioned expression formula can be reduced to:

${p^{\′}}_0=\frac{1}{2}{C}_p(2\stackrel{\‾}{\mathrm{\ρ}}U{u}^{\′}+{\mathrm{\ρ}}^{\′}{U}^2)$

$\frac{{p^{\′}}_0}{U^2}=\frac{1}{2}{C}_p(\frac{2\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′}}{U}+{\mathrm{\ρ}}^{\′})---\left(3\right)$

In formula (3), $U=\stackrel{\‾}{M}c=\stackrel{\‾}{M}\sqrt{\mathrm{\γR}\stackrel{\‾}{T}},\stackrel{\‾}{p_0}=\stackrel{\‾}{\mathrm{\ρ}}R\stackrel{\‾}{T},$ Wherein R is gas law constant.

2) set up the funtcional relationship between velocity fluctuation u' and density are ρ ';

Step 2 in step 2), in, the described concrete grammar of setting up the funtcional relationship between velocity fluctuation u' and density are ρ ' can be:

According to strong Reynolds analogy (SRA) relation proposing in Morkovin hypothesis, have following relational expression (P.BRADSHAW.Theeffect of mean compression or dilatation on the turbulence structure of supersonicboundary layers[J] .J.Fluid Mech.1974,63 (3): 449-464; M.PINO MARTIN.Directnumerical simulation of hypersonic turbulent boundary layers.Part1.Initializationand comparison with experiments[J] .J.Fluid Mech.2007,570:347-364):

$\frac{{\mathrm{\ρ}}^{\′}}{\stackrel{\‾}{\mathrm{\ρ}}}=(\mathrm{\γ}-1){\tilde{M}}^2\frac{u^{\′}}{U}---\left(4\right)$

Wherein be local mach number (or local Mach number), its expression formula is

$\tilde{M}=\frac{\tilde{u}}{\tilde{c}}$

Wherein represent the velocity of sound in high-speed wind tunnel, be the Density Weighted mean value of speed, u is carried out to Density Weighted decomposition, u can be expressed as wherein expression formula is u " be 1 speed trace (SergioPirozzoli; Francesco Grasso.Direct numerical simulation of impinging shockwave/turbulent boundary layer interaction at M=2.25[J] .PHYSICS OF FLUIDS (18); 065113,2006).Will formula substitution formula (4) can obtain

$\frac{{\mathrm{\ρ}}^{\′}}{\stackrel{\‾}{\mathrm{\ρ}}}=(\mathrm{\γ}-1){\tilde{M}}^2\frac{u^{\′}}{U}=(\mathrm{\γ}-1){\left(\frac{U+u^{\′}-u^{\′\′}}{\tilde{c}}\right)}^2\frac{u^{\′}}{U}$

Remove the second order a small amount of in above-mentioned equation, namely

$\tilde{M}={\left(\frac{U+u^{\′}-u^{\′\′}}{\tilde{c}}\right)}^2\≈{\left(\frac{U}{\tilde{c}}\right)}^2={\stackrel{\‾}{M}}^2$

Thereby can obtain the functional relation between following velocity fluctuation u' and density are ρ '

$\frac{{\mathrm{\ρ}}^{\′}}{\stackrel{\‾}{\mathrm{\ρ}}}=(\mathrm{\γ}-1){\stackrel{\‾}{M}}^2\frac{u^{\′}}{U}---\left(5\right)$

3) derive and obtain pressure fluctuation p' ₀and the functional relation between velocity fluctuation u', concrete grammar is as follows:

By in formula (5) substitution formula (3) to reach the object of eliminating density fluctuation, thereby obtain following equation:

$\frac{{p^{\′}}_0}{U^2}=\frac{1}{2}(\frac{2\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′}}{U}+(\mathrm{\γ}-1){\stackrel{\‾}{\mathrm{\ρM}}}^2\frac{u^{\′}}{U})$

Above-mentioned equation is converted and can obtain pressure fluctuation p' ₀functional relation with velocity fluctuation u':

$\frac{u^{\′}}{U}=\frac{2{p^{\′}}_0}{C_p\stackrel{\‾}{\mathrm{\ρ}}{U}^2[2+(\mathrm{\γ}-1){\stackrel{\‾}{M}}^2]}---\left(6\right)$

In above formula, except pressure fluctuation p' ₀outside two unknown numbers of velocity fluctuation u', other data all can record or be known, as long as therefore record pressure fluctuation p' ₀can calculate the value of velocity fluctuation u'; Computing formula because of turbulivity is again be the amount being proportionate with velocity fluctuation, therefore know that pressure fluctuation can calculate the value of turbulivity; Formula (6) is the functional relation of turbulivity and fluctuation pressure; By the fluctuation pressure value substitution formula (6) of each measuring point of wind-tunnel outlet, obtain the turbulivity of each measuring point.

Technical scheme of the present invention is comprised of two parts: the one, and wind tunnel data gatherer process, the realization of this process is to utilize the adjustable apparatus that miniature pitot is housed to measure the fluctuation pressure value of wind-tunnel incoming flow diverse location, and records other wind tunnel datas with other probes.The 2nd, data analysing method, the method is that pressure pulse value to having recorded and other wind tunnel datas are analyzed, and derives and obtains the functional relation of pressure fluctuation and velocity fluctuation in hypersonic air-flow, thereby calculate the turbulivity of hypersonic wind tunnel.

In the present invention, being used for measuring the miniature pitot of average stagnation pressure and pressure fluctuation, is to select the silicon pressure sensor kulite XCS-062 that can adapt to high-temperature high-frequency environment.Such pitot windward diameter of section only has 1.6mm, little with air-flow contact area, so the impact of excited wave is less, can adapt to the severe condition in hypersonic wind tunnel.

For controlling planar mobile adjustable apparatus of pitot, by drive vertical mobile support saddle vertically mobile screw mandrel, vertically mobile support saddle, drive the screw mandrel that moves horizontally bearing and move horizontally, move horizontally bearing, control the guide pole that moves left and right, with mechanisms such as two helical gear gear shafts, horizontal mobile mechanism outer cover, probe bearing, servomotors, form.

The present invention has following beneficial effect:

The present invention is by setting up the funtcional relationship between velocity fluctuation and pressure fluctuation, the measurement of velocity fluctuation is converted into the measurement of pressure fluctuation, use miniature high-frequency high temperature silicon pressure sensor, thereby can adapt to the severe condition of hypersonic wind tunnel, obtain measurement result comparatively reliably.In setting up the process of functional relation between velocity fluctuation and pressure fluctuation, ignore second order pulsation and the less factor of some other impact, thereby functional relation is simplified, use strong Reynolds analogy (SRA) relation of Morkovin hypothesis, utilize in compressible hypersonic speed flow and flow similar statistical average feature to incompressible hypersonic, finally set up the linear functional relation formula between pressure fluctuation and velocity fluctuation.Functional relation between pressure fluctuation and velocity fluctuation is simple and clear, only need record pressure fluctuation and just can obtain the turbulivity of hypersonic wind tunnel through simple computation, convenient and swift.

Accompanying drawing explanation

Fig. 1 controls the planar three-dimensional structure sketch of mobile adjustable apparatus of probe.

Fig. 2 is the cut-open view of adjustable apparatus horizontal mobile mechanism.

Fig. 3 is the three-dimensional plot that moves horizontally bearing and probe distribution.

Fig. 4 is virtual coordinate system and the measuring point distribution schematic diagram that move in probe bearing apparent wind hole outlet cross section.

Being labeled as in figure: 1 represents hypersonic wind tunnel, 2 represent vertical screw mandrel (for driving vertical mobile support saddle vertically to move), 3 represent probe bearing, 4 represent vertical mobile support saddle, 5 represent horizontal mobile mechanism outer cover, 6 represent gear shaft (with two spiral gears), 7 represent servomotor (for driving bearing level and vertically mobile), 8 represent probe (kulite sensor is housed), 9 represent to move horizontally bearing, 10 represent horizontal lead screw (drive moves horizontally bearing and moves horizontally), 11 represent guide pole, 12 represent wind-tunnel outlet measuring point, 13 represent the virtual coordinate system that move in probe bearing apparent wind hole outlet cross section.

Embodiment

Below in conjunction with accompanying drawing, enforcement of the present invention is elaborated.

With reference to Fig. 1, the preferred embodiments of the present invention provide a kind of adjustable apparatus that probe moves in two dimensional surface of controlling.Gear shaft 6 is connected with two vertical screw mandrels 2 respectively by two spiral gears, and motor 7 gear shaft 6 are rotated, and gear shaft 6 drives two vertical screw mandrels 2 to rotate by spiral gear.Between two vertical screw mandrels, horizontal mobile mechanism is housed, with reference to Fig. 2, horizontal mobile mechanism by two vertical mobile support saddles 4, horizontal mobile mechanism outer cover 5, move horizontally bearing 9, horizontal lead screw 10, guide pole 11 and form.Vertically mobile support saddle 4 is connected with vertical screw mandrel 2, and vertical mobile screw mandrel 2 rotations can drive vertical mobile support saddle 4 to move up and down.Move horizontally bearing 9 and be arranged on horizontal lead screw 10 and guide pole 11, horizontal lead screw 10 is driven and is rotated by motor 7, and drives and move horizontally bearing 9 and move left and right, and guide pole 11 is for guaranteeing to move horizontally moving horizontally of bearing 9.The design of the wedge angle of horizontal mobile mechanism outer cover 5 can make air-flow produce oblique shock wave backward, can prevent that perpendicular shock affects near the measurement flow field of probe 8.The distribution of probe 8 is with reference to Fig. 3, the installation direction of probe 8 flows to parallel with hypersonic wind tunnel 1, probe bearing 3 is crosswise, 5 probes 8 also become cross distribution on probe bearing 3, when probe bearing 3 moves to certain while a bit measuring, 5 probes 8 can measure the fluctuation pressure value of 5 coordinate points simultaneously.With reference to Fig. 4, the initial point of virtual coordinate system is seated in the center of circle of wind-tunnel outlet, and measuring point is uniformly distributed in outlet.

Embodiment: hypersonic wind tunnel turbulivity indirect measurement method, the wind-tunnel exit diameter that the present embodiment is given is 600mm, 4 probes of probe bearing coboundary are 25mm to the distance of probe, and on wind-tunnel outlet, measuring point all distributes with 25mm spacing at x axle and y direction of principal axis.Make measuring point coordinate on initial point for (0,0), toward x axle move x * 25mm measuring point coordinate be designated as (x, 0); Toward y axle move y * 25mm measuring point coordinate be designated as (0, y); If past x axle and y direction of principal axis simultaneously mobile measuring point coordinate are designated as (x, y).

Traveling probe bearing is measured different measuring points, because have 5 probes on probe bearing, therefore all can record the pressure pulse value of 5 different measuring points at every turn, the pressure fluctuation subscript that 1 measuring point is measured for the 1st time is labeled as 1, the 2nd time measured value subscript is labeled as 2, the like, the multipotency of each measuring point records five groups of data.First probe bearing center line is aimed to initial point, now record (0,0), (1,0), (0,1), (1,0), (0,-1) pressure fluctuation of 5 coordinate points, because the data of each measuring point are for the first time, record, the subscript of each data is 1, at this, 5 pressure pulse values that record is designated as to (p' ₁(0,0), p' ₁(1,0), p' ₁(0,1), p' ₁(1,0), p' ₁(0 ,-1)), then starter motor 7 drives horizontal lead screws 10 to drive the horizontal seats 25mm that moves right to arrive the 2nd measuring point (1,0), now records (1,0), (2,0), (1,1), (0,0), the pressure fluctuation (p' of (1 ,-1) 5 measuring points ₂(1,0), p' ₁(2,0), p' ₁(1,1), p' ₂(0,0), p' ₁(1 ,-1)), if after this will measure measuring point (1,1) pressure pulse value, starter motor 7 drives vertical mobile screw mandrel 2 to drive vertical mobile support saddle 4 to move 25mm to y axle forward and arrives measuring point (1,1), now record (1,1), (2,1), (1,2), (0,1), the pressure pulse value of (1,0) 5 measuring points is:

(p' ₂(1,1),p' ₁(2,1),p' ₁(1,2),p' ₂(0,1),p' ₃(1,0))

The measurement of other measuring points, until the pressure fluctuation of all measuring points all measured, mean pressure pulsating quantity p' on any 1 measuring point (x, y) ₀for

$p^{\′}(x,y)=\frac{\underset{i=1}{\overset{j}{\mathrm{\Σ}}}{p^{\′}}_i(x,y)}{5},1\≤j\≤5$

In above formula, i represents the measured value that measuring point is measured for the i time, and j represents measured j time of each measuring point, and its number of times maximum is no more than 5 times.In like manner can record the average stagnation pressure of any 1 measuring point

By being distributed near wind-tunnel outlet 4 differential pressure pickups and the thermopair on wall, record the static pressure of wind-tunnel outlet and stagnation temperature respectively it is averaged to obtain to the average static pressure of wind-tunnel outlet and medial temperature be respectively:

$\stackrel{\‾}{p}=\frac{\stackrel{\‾}{p_1}+\stackrel{\‾}{p_2}+\stackrel{\‾}{p_3}+\stackrel{\‾}{p_4}}{4}$

$\stackrel{\‾}{T}=\frac{\stackrel{\‾}{T_1}+\stackrel{\‾}{T_2}+\stackrel{\‾}{T_3}+\stackrel{\‾}{T_4}}{4}$

According to with can calculate U, c _pwith again by these data substitution formulas (6):

$\frac{u^{\′}(x,y)}{U}=\frac{2p^{\′}(x,y)}{C_p\stackrel{\‾}{\mathrm{\ρ}}{U}^2[2+(\mathrm{\γ}-1){\stackrel{\‾}{M}}^2]}$

Thereby calculate velocity fluctuation u'(x, y on measuring point (x, y)) and turbulivity thereby completing the turbulivity of arbitrfary point in wind-tunnel measures indirectly.

Claims (1)

1. a hypersonic wind tunnel turbulivity indirect measurement method, is characterized in that comprising the following steps:

One, gather wind tunnel data, concrete grammar is as follows:

1) control motor and make pitot aim at wind-tunnel export center, record the average stagnation pressure of the 1st measuring point with fluctuation pressure p' ₀;

2), by the horizontal or vertical Moving Unit length of pitot, record the pressure data of the 2nd measuring point;

3) continuous repeating step 2), record third and fourth ... the data of individual measuring point, until record all measuring point pressure datas that are evenly distributed on wind-tunnel outlet;

4) near wind-tunnel outlet, along wall, be circumferentially evenly arranged 4 differential pressure pickup and 4 thermopairs of measuring stagnation temperatures of measuring static pressure, record 4 static pressures and 4 stagnation temperature values are averaged and can obtain average static pressure and medial temperature

Two, the analysis to wind tunnel data, concrete grammar is as follows:

1) build-up pressure pulsation p' ₀and the funtcional relationship between velocity fluctuation u', density are ρ ', concrete grammar is:

In known wind-tunnel, the stagnation pressure expression formula of air-flow is:

$p_0=p+\frac{1}{2}{C}_p{\mathrm{\ρu}}^2---\left(1\right)$

P wherein ₀be stagnation pressure, p is static pressure, represent dynamic pressure, C _pbe pressure coefficient, ρ is the density of air, and u is the speed of air; Pressure coefficient C _pbe the amount relevant with Mach number M to air specific heat ratio γ, its expression formula is:

$C_p=\frac{4}{\mathrm{\γ}+1}(1-\frac{1}{M^2})---\left(2\right)$

By argument table be shown mean value and percent ripple and, even

$p_0=\stackrel{\‾}{p_0}+{p^{\′}}_0;p=\stackrel{\‾}{p}+p^{\′};M=\stackrel{\‾}{M}+M^{\′};\mathrm{\ρ}=\stackrel{\‾}{\mathrm{\ρ}}+{\mathrm{\ρ}}^{\′};u=U+u^{\′}$

Wherein be average static pressure, p' is pulsation static pressure, be average Mach number, M' is Mach number pulsation, be average density, U is average velocity, brings the expression formula of M into formula (2), has

$C_p=\frac{4}{\mathrm{\γ}+1}(1-\frac{1}{{(\stackrel{\‾}{M}+M^{\′})}^2})$

Due under hypersonic m' is very little, and under this functional relation, M' is to C _pimpact very little, therefore ignored, thus pressure coefficient C _pcan be expressed as

$C_p=\frac{4}{\mathrm{\γ}+1}(1-\frac{1}{{\stackrel{\‾}{M}}^2})$

In above formula, air specific heat ratio γ generally gets 1.40, average Mach number there is following relational expression:

$\frac{\stackrel{\‾}{p}}{\stackrel{\‾}{p_0}}={(1+\frac{\mathrm{\γ}-1}{2}{\stackrel{\‾}{M}}^2)}^{-\frac{\mathrm{\γ}}{\mathrm{\γ}-1}}$

Average static pressure in above formula average stagnation pressure therefore be known with air specific heat ratio γ, can calculate thereby can be derived from pressure coefficient C _pvalue;

By each argument table in formula (1) be shown mean value and percent ripple and, formula (1) can be expressed as

$\stackrel{\‾}{p_0}+{p^{\′}}_0=\stackrel{\‾}{p}+p^{\′}+\frac{1}{2}{C}_p(\stackrel{\‾}{\mathrm{\ρ}}{U}^2+\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′2}+2\stackrel{\‾}{\mathrm{\ρ}}U{u}^{\′}+{\mathrm{\ρ}}^{\′}{u}^{\′2}+2{\mathrm{\ρ}}^{\′}U{u}^{\′})$

Due to

$\stackrel{\‾}{p_0}=\stackrel{\‾}{p}+\frac{1}{2}{C}_p\stackrel{\‾}{\mathrm{\ρ}}{\stackrel{\‾}{u}}^2$

Therefore fluctuation pressure can be expressed as:

${p^{\′}}_0=p^{\′}+\frac{1}{2}{C}_p(\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′2}+2\stackrel{\‾}{\mathrm{\ρ}}U{u}^{\′}+{\mathrm{\ρ}}^{\′}{U}^2+{\mathrm{\ρ}}^{\′}{u}^{\′2}+2{\mathrm{\ρ}}^{\′}U{u}^{\′})$

Consider that single order percent ripple is very little, second order percent ripple almost can be ignored, thereby above-mentioned expression formula can be reduced to:

${p^{\′}}_0=\frac{1}{2}{C}_p(2\stackrel{\‾}{\mathrm{\ρ}}U{u}^{\′}+{\mathrm{\ρ}}^{\′}{U}^2)$

$\frac{{p^{\′}}_0}{U^2}=\frac{1}{2}{C}_p(\frac{2\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′}}{U}+{\mathrm{\ρ}}^{\′})---\left(3\right)$

In formula (3), $U=\stackrel{\‾}{M}c=\stackrel{\‾}{M}\sqrt{\mathrm{\γR}\stackrel{\‾}{T}},\stackrel{\‾}{p_0}=\stackrel{\‾}{\mathrm{\ρ}}R\stackrel{\‾}{T},$ Wherein R is gas law constant;

2) set up the funtcional relationship between velocity fluctuation u' and density are ρ ', concrete grammar is:

Strong Reynolds analogy (SRA) relation according to proposing in Morkovin hypothesis, has following relational expression:

$\frac{{\mathrm{\ρ}}^{\′}}{\stackrel{\‾}{\mathrm{\ρ}}}=(\mathrm{\γ}-1){\tilde{M}}^2\frac{u^{\′}}{U}---\left(4\right)$

Wherein be local mach number or local Mach number, its expression formula is

$\tilde{M}=\frac{\tilde{u}}{\tilde{c}}$

Wherein represent the velocity of sound in high-speed wind tunnel, be the Density Weighted mean value of speed, u is carried out to Density Weighted decomposition, u can be expressed as wherein expression formula is u " is 1 speed trace; Will formula substitution formula (4) can obtain

$\frac{{\mathrm{\ρ}}^{\′}}{\stackrel{\‾}{\mathrm{\ρ}}}=(\mathrm{\γ}-1){\tilde{M}}^2\frac{u^{\′}}{U}=(\mathrm{\γ}-1){\left(\frac{U+u^{\′}-u^{\′\′}}{\tilde{c}}\right)}^2\frac{u^{\′}}{U}$

Remove the second order a small amount of in above-mentioned equation,

$\tilde{M}={\left(\frac{U+u^{\′}-u^{\′\′}}{\tilde{c}}\right)}^2\≈{\left(\frac{U}{\tilde{c}}\right)}^2={\stackrel{\‾}{M}}^2$

Thereby can obtain the functional relation between following velocity fluctuation u' and density are ρ '

$\frac{{\mathrm{\ρ}}^{\′}}{\stackrel{\‾}{\mathrm{\ρ}}}=(\mathrm{\γ}-1){\stackrel{\‾}{M}}^2\frac{u^{\′}}{U}---\left(5\right)$

3) derive and obtain pressure fluctuation p' ₀and the functional relation between velocity fluctuation u', concrete grammar is as follows:

By in formula (5) substitution formula (3) to reach the object of eliminating density fluctuation, thereby obtain following equation:

$\frac{{p^{\′}}_0}{U^2}=\frac{1}{2}(\frac{2\stackrel{\‾}{\mathrm{\ρ}}{u}^{\′}}{U}+(\mathrm{\γ}-1){\stackrel{\‾}{\mathrm{\ρM}}}^2\frac{u^{\′}}{U})$

Above-mentioned equation is converted and can obtain pressure fluctuation p' ₀functional relation with velocity fluctuation u':

$\frac{u^{\′}}{U}=\frac{2{p^{\′}}_0}{C_p\stackrel{\‾}{\mathrm{\ρ}}{U}^2[2+(\mathrm{\γ}-1){\stackrel{\‾}{M}}^2]}---\left(6\right)$

In above formula, except pressure fluctuation p' ₀outside two unknown numbers of velocity fluctuation u', other data all can record or be known, as long as therefore record pressure fluctuation p' ₀can calculate the value of velocity fluctuation u'; Computing formula because of turbulivity is again be the amount being proportionate with velocity fluctuation, therefore know that pressure fluctuation can calculate the value of turbulivity; Formula (6) is the functional relation of turbulivity and fluctuation pressure; By the fluctuation pressure value substitution formula (6) of each measuring point of wind-tunnel outlet, obtain the turbulivity of each measuring point.