CN103969022B - A kind of hypersonic wind tunnel turbulivity indirect measurement method - Google Patents

Abstract

A kind of hypersonic wind tunnel turbulivity indirect measurement method, relates to Wind Tunnel Turbulence Spectral Analyzer mobility and measures.There is provided and utilize miniature pitot to measure pressure fluctuation in hypersonic wind tunnel flow field, then convert according to the relation of pressure fluctuation and velocity fluctuation and indirectly obtain a kind of hypersonic wind tunnel turbulivity indirect measurement method of turbulivity.Comprise wind tunnel data collection and data analysis, the process of wind tunnel data collection is the fluctuation pressure value utilizing the adjustable apparatus that miniature pitot is housed to measure wind-tunnel incoming flow diverse location, and records other wind tunnel datas with other probes.The method of data analysis analyzes the pressure pulse value recorded and other wind tunnel datas, derives and obtain the functional relation of pressure fluctuation and velocity fluctuation in hypersonic air-flow, thus calculate the turbulivity of hypersonic wind tunnel.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.

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 find, turbulivity to boundary layer transition important, its on the impact of pneumatic gauging seriously can compared with Reynolds number.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 before emulation or estimate, the accuracy of turbulence measurement value or estimated value directly affects the accuracy of simulation result.

At present, the measurement of turbulivity generally uses hot-wire anemometer (HWA), Laser Doppler Velocimeter (LDV) and particle image speed-measuring system (PIV), and they are all by the velocity fluctuation in measurement flow field and then calculate turbulivity.The ultimate principle of HWA is heat balance principle, the fine wire with heating current is placed in flow field, the change of wind speed can make temperature wiry change, by measuring the voltage at hot line two ends thus flow velocity (E.L.DOUGHMAN.DevelopmentofaHot-WireAnemometerforHyperson icTurbulentFlows [J] .THEREVIEWOFSCIENTIFICINSTRUMENTS.VOLUME43 can being calculated, NUMBER8,1972).The ultimate principle of LDV measures the Doppler signal of the trace particle by laser probe, and then converting according to the relation of speed and Doppler frequency obtains speed.PIV technology is based upon on the basis of Flow visualisation technology, its ultimate principle is in flow field, add certain trace particle, the test section in the illuminated with laser light flow field sent with pulsed laser, and the image of fluidized particle between test section is taken by CCD camera, velocity distribution (the PenttiSaarenrinne in flow field is obtained after analyzing and processing is carried out to image, MikaPiirtoandHannuEloranta.Experiencesofturbulencemeasur ementwithPIV [J] .MEASUREMENTSCIENCEANDTECHNOLOGY, 2001).

At present, yet there are no both at home and abroad about can the relevant report of method of Measurement accuracy hypersonic wind tunnel turbulivity.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 the turbulence measurement be not suitable in Hypersonic Flow Field.It is high that HWA has dynamic response frequency, Time and place 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 belongs to noncontact Flow Visualization Technologies, and their measuring accuracy are high, and the scope that tests the speed is wide, but in High Speed Flow Field, there is trace particle followability poor, and excited wave affects large shortcoming.

Summary of the invention

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

The present invention includes following steps:

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

1) controlling motor makes pitot aim at wind-tunnel export center, records the average total pressure of the 1st measuring point with fluctuation pressure p' ₀;

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

3) constantly step 2 is repeated), record third and fourth ... the data of individual measuring point, until record all measuring point pressure datas being evenly distributed on wind-tunnel outlet;

4) near wind-tunnel outlet, be evenly arranged 4 along wall circumference and measure the differential pressure pickup of static pressure and the thermopair of 4 measurement stagnation temperatures, 4 static pressures recorded and 4 total temperature value are averaged and can obtain average static pressure and medial temperature ;

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

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

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

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

Wherein p 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:

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

Variable is expressed as 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, the expression formula of M is brought into formula (2), then 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 to be 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 total pressure be known with air specific heat ratio γ, therefore can calculate thus can be derived from pressure coefficient C _pvalue;

Each variable in formula (1) is expressed as 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, and thus 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) funtcional relationship between velocity fluctuation u' and density ρ ' is set up;

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

According to strong Reynolds analogy (SRA) relation proposed in Morkovin hypothesis, there are following relational expression (P.BRADSHAW.Theeffectofmeancompressionordilatationonthetu rbulencestructureofsupersonicboundarylayers [J] .J.FluidMech.1974,63 (3): 449-464; M.PINOMARTIN.Directnumericalsimulationofhypersonicturbul entboundarylayers.Part1.Initializationandcomparisonwithe xperiments [J] .J.FluidMech.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, carry out Density Weighted decomposition to u, then u can be expressed as wherein expression formula is u " is 1 speed trace (SergioPirozzoli; FrancescoGrasso.Directnumericalsimulationofimpingingshoc kwave/turbulentboundarylayerinteractionatM=2.25 [J] .PHYSICSOFFLUIDS (18); 065113,2006).Will formula substitute into 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

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

Thus the functional relation that can obtain between following velocity fluctuation u' and density ρ '

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

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

Formula (5) is substituted into reach the object eliminating density fluctuation in formula (3), thus obtains following equation:

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

Conversion is carried out to above-mentioned equation and can obtain pressure fluctuation p' ₀with the functional relation of 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 velocity fluctuation u' two unknown numbers, other data all can record or be known, as long as therefore record pressure fluctuation p' ₀the value of velocity fluctuation u' can be calculated; Again because the computing formula of turbulivity is be the amount be 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; The fluctuation pressure value of each for wind-tunnel outlet measuring point is substituted into formula (6), namely obtains the turbulivity of each measuring point.

Technical scheme of the present invention is made up of two parts: one is wind tunnel data gatherer process, and the realization of this process is the fluctuation pressure value utilizing the adjustable apparatus that miniature pitot is housed to measure wind-tunnel incoming flow diverse location, and records other wind tunnel datas with other probes.Two is data analysing methods, and the method analyzes the pressure pulse value recorded and other wind tunnel datas, derives and obtains the functional relation of pressure fluctuation and velocity fluctuation in hypersonic air-flow, thus calculate the turbulivity of hypersonic wind tunnel.

Being used for measuring the miniature pitot of average total pressure and pressure fluctuation in the present invention, is select the silicon pressure sensor kuliteXCS-062 that can adapt to high-temperature high-frequency environment.Such pitot windward diameter of section only has 1.6mm, little with air flow contacts area, and therefore the impact of excited wave is less, can adapt to the harsh conditions in hypersonic wind tunnel.

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

The present invention has following beneficial effect:

The present invention is the funtcional relationship by setting up 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, thus the harsh conditions of hypersonic wind tunnel can be adapted to, obtain measurement result comparatively reliably.In the process setting up functional relation between velocity fluctuation and pressure fluctuation, ignore the factor that second order is pulsed and some other impact is less, thus functional relation is simplified, use strong Reynolds analogy (SRA) relation of Morkovin hypothesis, utilize statistical average feature similar to incompressible hypersonic stream in compressible hypersonic speed flow, 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 is the three-dimensional structure sketch controlling the probe planar adjustable apparatus of movement.

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

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

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

Being labeled as in figure: 1 represents hypersonic wind tunnel, 2 represent vertical screw mandrel (vertically moving for driving vertical mobile support saddle), 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 moving), 8 represent probe (kulite sensor is housed), 9 expressions 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, the virtual coordinate system of 13 expression probe bearing apparent wind hole outlet cross section movements.

Embodiment

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

With reference to Fig. 1, the preferred embodiments of the present invention provide a kind of adjustable apparatus controlling probe movement in two dimensional surface.Gear shaft 6 by two spiral gears respectively screw mandrel 2 vertical with two be connected, motor 7 gear shaft 6 is 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 form.Vertical mobile support saddle 4 is connected with vertical screw mandrel 2, and the rotation of vertical mobile screw mandrel 2 can drive vertical mobile support saddle 4 to move up and down.Move horizontally bearing 9 to be arranged on horizontal lead screw 10 and guide pole 11, horizontal lead screw 10 drives rotation by motor 7, and drives and move horizontally bearing 9 and move left and right, and guide pole 11 moves horizontally moving horizontally of bearing 9 for guaranteeing.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 the measurement flow field near 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 in crosswise, 5 probes 8 also become cross distribution on probe bearing 3, when probe bearing 3 move to certain a bit measure time, 5 probes 8 can measure the fluctuation pressure value obtaining 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, probe bearing coboundary 4 probes are 25mm to the distance of probe, and on wind-tunnel outlet, measuring point all distributes with 25mm spacing in x-axis and y-axis direction.Make measuring point coordinate on initial point for (0,0), toward x-axis move x × 25mm then measuring point coordinate be designated as (x, 0); Toward y-axis move y × 25mm then measuring point coordinate be designated as (0, y); If move toward x-axis and y-axis direction simultaneously, measuring point coordinate is designated as (x, y).

Traveling probe bearing is measured different measuring points, because probe bearing has 5 probes, therefore all can record the pressure pulse value of 5 different measuring points at every turn, 1 measuring point is measured the pressure fluctuation subscript obtained for the 1st time and is labeled as 1,2nd time measured value subscript is labeled as 2, the like, the most multipotency of each measuring point records five groups of data.First probe bearing center line is aimed at initial point, now record (0,0), (1,0), (0,1), (-1,0), (0,-1) pressure fluctuation of 5 coordinate points, data because of each measuring point are and record for the first time, and the subscript of each data is 1, at this, record 5 pressure pulse values is designated as (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 screw 10 to drive the horizontal seat 25mm that moves right to arrive the 2nd measuring point (1,0), now record (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)), after this to measure measuring point (1,1) pressure pulse value, then starter motor 7 drives vertical mobile screw mandrel 2 to drive vertical mobile support saddle 4 to move 25mm to y-axis 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 the like, until the pressure fluctuation of all measuring points is all measured, the 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 i-th time, and j represents measured j time of each measuring point, and its number of times is maximum is no more than 5 times.In like manner can record the average total pressure of any 1 measuring point

By being distributed in 4 differential pressure pickups near wind-tunnel outlet on wall and thermopair, record the static pressure of wind-tunnel outlet and stagnation temperature respectively to its be averaged wind-tunnel outlet average static pressure 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 these data are substituted into formula (6):

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

Thus the velocity fluctuation u'(x, the y that calculate on measuring point (x, y)) and turbulivity thus complete the turbulivity indirect inspection of arbitrfary point in wind-tunnel.

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) controlling motor makes pitot aim at wind-tunnel export center, records the average total pressure of the 1st measuring point with fluctuation pressure p' ₀;

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

3) constantly step 2 is repeated), record third and fourth ... the data of individual measuring point, until record all measuring point pressure datas being evenly distributed on wind-tunnel outlet;

4) near wind-tunnel outlet, be evenly arranged 4 along wall circumference and measure the differential pressure pickup of static pressure and the thermopair of 4 measurement stagnation temperatures, 4 static pressures recorded and 4 total temperature value are averaged and can obtain average static pressure and medial temperature

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

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

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

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

Wherein p 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:

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

Variable is expressed as 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, the expression formula of M is brought into formula (2), then 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 to be 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 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 total pressure be known with air specific heat ratio γ, therefore can calculate thus can be derived from pressure coefficient C _pvalue;

Each variable in formula (1) is expressed as mean value and percent ripple and, formula (1) can be expressed as

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

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:

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

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

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

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

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

According to strong Reynolds analogy (SRA) relation proposed in Morkovin hypothesis, there is 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, carry out Density Weighted decomposition to u, then u can be expressed as wherein expression formula is u " is 1 speed trace; Will formula substitute into 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

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

Thus the functional relation that can obtain between following velocity fluctuation u' and density ρ '

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

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

Formula (5) is substituted into reach the object eliminating density in formula (3), thus obtains following equation:

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

Conversion is carried out to above-mentioned equation and can obtain pressure fluctuation p' ₀with the functional relation of velocity fluctuation u':

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

In above formula, except pressure fluctuation p' ₀outside velocity fluctuation u' two unknown numbers, other data all can record or be known, as long as therefore record pressure fluctuation p' ₀the value of velocity fluctuation u' can be calculated; Again because the computing formula of turbulivity is be the amount be 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; The fluctuation pressure value of each for wind-tunnel outlet measuring point is substituted into formula (6), namely obtains the turbulivity of each measuring point.