Rough Surface Aerodynamic Computation in Rarefied Gas Flow Applying the Solution of Inverse Problem

Most effective method to find the roughness parameters in rarefied gas flow is to calculate them from aerodynamic measurements, solving the inverse problem. The value of the main roughness parameter obtained from the solution of inverse problem is substantially higher (at least 1,25–1,5 times) than similar value of the same parameter measured from the profile diagrams. Thus, the effect of surface roughness in aerodynamic values of rough surface in rarefied gas flow is always significantly underestimated. First main reason of it is the low precision of roughness parameter measurements from the profile diagrams, and the second is based on usual lack of taking into account aerodynamic shadowing effect.


Introduction
Aerodynamic experiments in rarefied gas and accorfing calculations underestimate surface roughness.The method of numerical computation of the influence of surface roughness on aerodynamical characteristics in rarefied gas flow built on obtained solution of the inverse problem is proposed in the present paper.The inverse problem means here that surface roughness parameters are determined from aerodynamic experimental data.
Surface roughness is qualified usually in the technics by means of root-mean-square variation of a rough surface σ = (M z 2 ) 1/2 , where z(x, y) is an applicate of random field (which is the model of the roughness), i. e. the coordinate orthogonal to its mean level.Similar parameters of surface roughness are the average distance R z between the maximal and the minimal values of z-coordinate, and the distance between the peaks.However, describing surface roughness in rarefied gas flow we need the parameter σ 1 = (M z 2 x ) 1/2 = (M z 2 y ) 1/2 determined not by surface height z, but by its partial derivatives z x = ∂z/∂x and z x = ∂z/∂y in directions x and y of the mean level of a random field [1]- [6].But the technical standard of surface quality does not contain the value σ 1 .Conquently, to find the parameter σ 1 in the papers becomes a complicated problem, and the measurements of σ 1 on the profile diagrams are extremely inaccurate [1].Thus, retrieveing the roughness parameter σ 1 from aerodynamic experiments solving the inverse problem remains the only exact algorithm to determine its real values.Analytical and numerical study of the influence of surface roughness on scattering function and on momentum and energy exchange coefficients in rarefied gas flow is applied to solve the inverse problem.[5], [6], and generalized to poly-Gaussian model of the roughness in [4].Both models have several advantages in comparison with other different more simple models of surface roughness applied in practice, including fractal roughness [5], [11], polygonalline model [7], and the surfaces with conical holes [8], and constructed of flat elements [9]- [10].In particular, the disadvantage of these simple models is accounting only of large-scale roughness input into momentum and energy exchange coefficients on the surface [5].All these models are too simple, therefore they could not take into account the micro-roughness (the roughness of smaller scale) -only poly-Gaussian model is complicated enough to describe it exatly.Moreover, this model has been approved on the base of experimental data [12].
The preference of poly-Gaussian model of roughness to Gaussian random field in different applications is justified in [12] and in [4]: "Non-Gaussian statistics have rough surfaces of mixed structure, modelled as a result of several stages of processing (by pressing, extrusion, grinding, honing etc.) of wearing surfaces after being operated, maintained and so on.Other reason for non-Gaussian relief is the deterministic component, which is typical for the processing by whetting, milling, polishing etc." The statement of the problem is based on the assumption that the number of rough irregularities in a small surface area dS is large.Therefore, the scale of these irregularities, and consequently, the characteristic scale of the roughness is smaller than the characteristic scale of the flow.Only this assumption permits to separate roughness operator Ŝ and local scattering function V 0 in the basic expression V = ŜV 0 .Here the operator Ŝ describes the influence of geometrical shape of a roughness (and of the trajectory of a reflected gas atom) on V .Consequently, physical and chemical parameters of the gas and of the surface have no effect on Ŝ.These parameters of the gas and of the surface influence only the local scattering function V 0 (scattering function on plane surface without roughness).
The scattering function V (v, v ) = Ŝ1 V 0 (v, v , n) on the rough surface takes the form [5] V Here V 0 (v, v , n) is the local scattering function, v and v are the orts of the velocities of incident and reflected gas atoms, v = (0, − sin θ, − cos θ), M 1 is the normalizing factor [3], and only the first reflections from the surface are considered, if there are multiple collisions.Scalar product of the vectors n and v is denoted by (n • v ); n and N are local and global normal unit vectors to the rough surface (n -normal to the smaller area on a rough surface), N and z-axis are directed perpendicular to the mean level of the roughness (Fig. 1).
The expression (1) of scattering function contains two roughness parameters: -the square mean derivative σ 1 ; -functional roughness parameter: the normalized correlation function ρ of the random field z(x, y).
The values σ 1 and ρ are most important for gas interaction, because all the other statistical parameters of the roughness (σ, R z , skewness coefficient etc.) have less influence on aerodynamics than the main rough surface parameters σ 1 and ρ [5].
The function p(z, z x , z y ) in ( 1) is poly-Gaussian probability density x 0 ,y 0 (z, z x , z y ) dF (v). ( Figure 1.First reflection of a gas atom with incident velocity v from smaller area with local normal n on a rough surface. where z(x, y), z x = ∂z/∂x and z y = ∂z/∂y are random field and its derivatives in the point (x 0 , y 0 ) of a gas atom scattering from the surface, and g v 2 r x 0 ,y 0 (z, z x , z y ) is corresponding Gaussian density with correlation function v 2 ρ.The spectrum of correlation function ρ contains necessarily the continuous component [3], and weighted function F (v) must be non-decreasing on the interval [0; ∞).

Aerodynamic shadowing effect in rarefied gas
Applying the expression of scattering function (1) to compute aerodynamic characteristic of rarefied gas flow near rough surface, we have trouble computing the conditional probability Π (1)  by given z, z x , z y Π (1) (θ, σ 1 ; z, z x , z y ) = P {z(0, y) ≤ z + y cot θ|z(0, 0) = z, z x (0, 0) = z x , z y (0, 0) = z y }, because Π (1) is the probability that the trajectory of the gas atom has no crossings with random field z(x, y), and the calculation of such probabilities is known complex promlem in the theory of random fields [1], [5], [6].
The factor Π (1) in ( 1) is important, because it takes into consideration significant effect: incident or reflected rarefied gas atoms could not reach substantial part of the area on a rough surface being blocked by emerging peaks of rough relief.Hence, these parts of the area are not to be accounted in rarefied gas-surface interaction.This effect is known as aerodynamic shadowing.Scattering function (1) could be computed far simpler without the multiplyer (3), as Π (1) depends on the whole implementation of a random field under gas atom trajectory, therefore, (3) is the integral in functional space known as continuum integral.Computation of the integrals of this type requires their approximation by the integrals of dimension n = 200 or more [5] Π (1) (4) Here (0, 0, z) is the point of the first scattering of a gas atom from the surface, and p is joined poly-Gaussian probability density of random field z(x, y) and its derivatives η 1 , η 2 , . . .η n in n points u 1 , . . ., u n , placed along the trajectory of incident atom.Considered points are distributed uniformly along the x-axis (x = 0, h, 2h, . . ., nh), and h is appropriate increment along the x-axis in the computation.
Applying the representation (4) in numerical calculation methods (like DSMC) in rarefied gas is inefficient even on most powerful computers.The complexity of considering aerodynamic shadowing effect and of practical usage of the calculation of the integrals of high dimension has resulted that this problem is studied only in St.-Petersburg.In many other papers (e.g. in [7]- [10]) aerodynamic shadowing is eliminated from the description of the interaction between the rarefied gas flow and the rough surface.
Computing the continuum integrals continues to be very complex not only if Gaussian or poly-Gaussian simulations of rough surface are used, but also for other models of the roughness.Taking into consideration aerodynamic shadowing on rough surface, we need to calculate continuum integrals (4) even for surface roughness constructed of randomly distributed flat elements.In this case p becomes joined probability density of correspondingly distributed random field z(x, y).Numerical investigation of rarefied gas-rough surface interaction without calculating continuum integrals becomes possible only for simplest deterministic roughness models like regular periodic functions (sine waves etc.).However, the approach based on deterministic models of roughness ignores the roughness of small scale, consequenly it underestimates the roughness effect in aerodynamic macro-parameters (reducing it to a half or even 3 times in practice).
The conditional probability (3) (or (4) -that the trajectory of the gas atom has no crossings with random field z(x, y)) depends on the implementation of the random field only under the trajectory of a gas particle.Hence, only the profile of the surface can be considered in the plane of two vectors: n and v (or v ).First is the local normal n to the surface, and second is the velocity of a gas atom.Therefore, we can study the properties of the probability Π (1) for poly-Gaussian random processes instead of more complicated probability for a random field.
Poly-Gaussian probability density of random process z(y) in n points y 1 , . . ., y n is the mixture of Gaussian densities: Here σ(v) is the mean of the vector z = (z 1 , . . ., z n ) of random process values in n points y 1 , . . ., y n , the transposition is denoted by upper index T , R n (v) is correlation matrix with determinant |R n (v)| and the inverse R −1 n (v), weighted function F (v) is continuous or discrete, and non-decreasing on [0; ∞).The integral ( 5) is supposed to be convergent.
In addition to discussed above merits of considered random fields, poly-Gaussian random process has two more known advantages: -all random processes could be fitted by poly-Gaussian one with any required precision [13].
-simple numerical algorithm based on the transformation of Gaussian distributions can be used to simulate sample relief of poly-Gaussian random process [12]; this algorithm is developed for different applications including surface diagnostics by electronic spectroscopy, light scattering on rough surface, growing of thin films for the micro-electronics, a friction in machinery and others [12].
Exact approximation of applied in practice micro-reliefs by poly-Gaussian random processes is approved also in many technological processes (e.g., rough surfaces are simulated made of the steel processed as well by ionic bombardment by nitrogen ions, as by chemical etching by alcohol solution of nitric acid -in both cases, experimental measurements confirmed numerical results with high precision).Similar techniques are used to process the surfaces of flying vehicles moving in upper atmosphere.

Applying Rice expansion in numerical calculations of rarefied gas atoms scattering from rough surface
Continuum integral Π (1) in ( 1)-( 4) can be computed expanding it in Rice series proposed by Miroshin [6] Π (1) Here N m are the factorial moments of m-th order of the number of the excursions of random field z(x, y) above the trajectories of gas atoms.The advantage of applying Rice series ( 6)) is the representation of factorial moments N m as the 2m-fold integrals of the probability density p (like in ( 4)).Rice series usually converges quickly, so the dimension of computed integrals reduces about 10 times -from 200 (or more) to 20-30.The representation of the factorial moments N m (u) of m-th order of the number A u [0; T ] of random field excursions above the level u of gas atom trajectory has been derived in [14] from the generalized expression (2) of poly-Gaussian probability density of random field and of its derivatives Here N G m are corresponding Gaussian factorial moments for divided by v level u v .Asymptotic evaluation of the values N m (u) in ( 6))-( 7)) is well-known for Gaussian processes in case of high level u [6].For poly-Gaussian processes we obtain from (7) similar evaluation lim ) of correlation function for h → 0 (α > 0).The consequence of the asymptotic evaluation of the factorial moments N m (u) is that the asymptotics of factorial moments (and, hence, of the number of poly-Gaussian process excursions above the high level) is similar to corresponding Gaussian asymptotics only if the function F (v) determines finite interval for the parameter v: F (v) = 1 for all v > v 0 , and lim It means that the mixture of Gaussian distributions defining poly-Gaussian field has Gaussian component with the maximal value v 0 of the variable v.In all other cases poly-Gaussian moments (7) are asymptotically significantly different from Gaussian moments.Thus, aerodynamic macro-parameters of poly-Gaussian random rough surface are usually substantially different from corresponding macro-parameters of Gaussian random rough surface.This result is quite consistent with difference between typical poly-Gaussian and Gaussian profiles and probability densities (5) as presented in [12].
Achieved results could be applied to study the dependence of aerodynamic macro-parameters of rough surface in rarefied gas flow on the roughness parameters (σ 1 and ρ) in two ways.
1. Known asymptotic evaluations on smooth surface (i.e. for σ 1 → 0 [6]) for Gaussian random field could be applied to derive the depence on σ 1 and ρ analytically.But similar asymptotics in Gaussian case is very complicated and corresponding evaluations contain continuum integrals (integrals in a functional space).To calculate them we need to approximate these integrals by the integrals of high dimensions (4).Moreover, the roughness of smaller scale (always present on real surfaces even by the best processing) does not allow to consider the surface as smooth.That is why the asymptotics on smooth surface is not used in practical calculations.However, one important result of asymptotic evaluations is that the dependence on surface roughness for poly-Gaussian model of roughness is significantly higher, than for Gaussian model: next factorial moment N m+1 is infinitely large of higher order, than previous one N m ).
2. It is possible to use poly-Gaussian model in numerical DSMC calculations.Herewith, accoding to previous conclusions, appropriate parameters of random field could be selected on preliminary stage of the computation.However, attention should be payd to the model of roughness: it must meet two stated criteria.
1) Its properties for simulating rarefied gas particles interacting with the surface (depending on physical and chemical characteristics of the gas and of the surface) must allow optimizing numerical procedures.
2) The coincidence is necessary between the roughness profiles of practically used surfaces and of simulating random field.

Inverse problem on rough surface in rarefied gas and the algorithm of its solution
To solve the inverse problem of determining the main surface roughness parameter from aerodynamic experimental data we need first to study the dependence of the scattering function V and of the momentum and energy exchange coefficients on roughness parameters σ 1 .
Expanding the local scattering function V 0 (v, v , n) (on a plane surface) in a series deriving general expansion of the scattering function V on a rough surface Here b k (v, v ) are the coefficients in the expansion ( 8), and continuum integrals are depending only on the parameters of random field simulating rough surface (9) The integrals K k (v, v ) are entirely determined by roughness parameters in rarefied gas (σ 1 and ρ(r)) and by the trajectory of gas particle.Therefore, these integrals could be computed preliminary, before the rarefied gas flow computation by DSMC method.The local scattering function V 0 (v, v , n) in this calculation could have an arbitrary analytical expression, but it must have an expansion in a series by system of the orthogonal functions ζ k (n) with only a few parameters to correspond to the physical and the chemical properties of the gas and of the surface.Applied in practical calculations scattering functions (e.g.diffuse, specular, Nochilla, Cercignani-Lampis and others) satisfy this restriction.Fig. 2 shows the dependence of computed integral (9) on roughness parameter σ 1 for diffuse-specular local scattering.The decreasing of the accuracy for σ 1 > 1/3 and θ > 70 • could be explained by insufficient number of the realizations of random field in computation.
The estimated coefficients b k (v, v ) in ( 8) can be transformed according to the roughness operator in (9) by given analytical local scattering model V 0 (v, v , n) (for the plane surface).The values K k (v, v ) (9) must be calculated at preliminary stage.According to the expressions (8)-( 9), the parameters on the rough surface are linear combinations of the parameters on the plane surface.The coefficients of these linear combinations depend on the values K k (v, v ).Simulating the calculated distribution, we obtain the velocities v of gas particles that are reflected from the rough surface.Described algorithm has an advantage in computational speed over the methods simulating the shape of the rough surface by simple geometrical models [7], [8], [9].This is the result of taking into consideration the entire shape information at the preliminary stage of computation.Consequently, the geometrical-shape simulation is excluded from the main DSMC procedure.From the consideration follows next algorithm of the solution of the inverse problem based on the calculation of surface roughness effect in rarefied gas flow.
1. Expanding the function V 0 in a series by the orthogonal functions ζ k (n) (trigonometric approximations applied in the local interaction theory have been applied in our calculations).
2. Computing the coefficients b k (v, v ) corresponding to the expansion of V 0 (v, v , n).
3. Calculating the values (9) in the functional space of the realizations of random field.It is necessary to determine the continuum integrals K k (v, v ) for various incidence angles, reflection angles, numbers k, and for various values of roughness parameter σ 1 .Then the obtained integrals K k (v, v ) in functional space should be expanded in a series in terms of the system ζ k (N) for different values of σ 1 and k (where the normal vector N to the average level of rough surface is directed along z-axis -fig.1). 4.
Expanding the scattering function V (v, v ) on rough surface received from the experimental data (or expanding the momentum p, τ and energy q exchange coefficients) in a series in terms of the system ζ k (N).
5. Calculating the continuum integrals K k (v, v ) from two expansions of V (v, v ) in a series in terms of orthogonal functions ζ k (N) with already known coefficients b k (v, v ).If the experimental values of V (v, v ) are here applied, then the solution of the inverse problem is achieved, finding as the roughness parameter σ 1 , as well the correlation function ρ(r) from calculated K k (v, v ).
6. Computing the scattering function V on a rough surface by ( 8)-( 9) gives finally the dependence of V (v, v ) on real surface roughness, obtained from the solution of the inverse problem.Substituting the expansion (8) into the integrals representing momentum p, τ and energy q exchange coefficients, we obtain the dependence of p, τ and energy q on surface roughness.

DSMC computation of rarefied gas atoms scattering from rough surface
The inverse problem should be solved on the first stage of the suggested algorithm on the base of the original experimental measurements of aerodynamic values (scattering function, momentum or energy exchange coefficients) on the surface of the same material with various roughness parameters.Experimental data from [9] and [10] is used by testing suggested algorithm for the solution of the inverse problem.
The results of this calculation for two different models of a surface roughness compared to experimental data from TSAGI (Central Aero Hydrodynamics Institute of Russia) [9]- [10] is presented in Fig. 3. Argon atoms with incidence angles θ = 0 • and θ = 60 • scattering from Kapton surface are studied.Solid lines demonstrate our results on poly-Gaussian surface with the same roughness parameters as in the numerical results received by Erofeev, Friedlander et.al.[9] (dash-dotted lines).Dotted line shows the indicatrix (projection of scattering function) on plane surface (without roughness); it coincides well with Nochilla and Cercignani-Lampis models, and is closer to diffuse scattering, than to specular or ray reflection.Different signs show experimental results [9] for Kapton surface.White and black signs (or first and second graph) specify smooth and rough surface respectively.It follows from the analysis of the graphs that either aerodynamic shadowing effect failed to take into account in [9], or considered partially: aerodynamic shadowing effect gives a wider indicatrix graph as well onwards, as in backwards direction.
The solution of the inverse problem by the suggested algorithm provides the value of the roughness parameter σ 1 1,1-1,3 times higher (depending on the local scattering kernel), than the value of σ 1 from measurements of geometrical characteristics of artificial roughness (or from optical measurements on the profile diagram).The difference between the results is the influence of the micro-roughness invisible on these diagrams.
Similar results are obtained from another experiments, e. g. from the normal momentum exchange coefficients of reflected argon and helium atoms with various molecular energy of incident molecular beam and various roughness of Kapton or aluminium surface (Fig. 4).The calculated value of σ 1 is 5% -35% higher in all calculations than in profile measurements.

Conclusion
1. Simulation of surface roughness by Gaussian and poly-Gaussian random fields improves the accuracy of calculating rarefied gas flow near rough surface.Necessary stage of the algorithm is analytical and numerical solution of the inverse problem of determining surface roughness parameter σ 1 in the interaction with rarefied gas flow.
2. The advantage of the solution of inverse problem is the possibility of exact accounting of the micro-roughness which could not be measured from the profile diagrams.Moreover, technical standard of surface quality does not contain the main roughness parameter σ 1 .
3. Poly-Gaussian model of roughness, applied to take into account aerodynamic shadowing effect on rough surface, permits constructing the most effective solution of inverse problem.Taking into account aerodynamic shadowing effect on rough surface (which is usually underestimated), we need to calculate functional space integrals in scattering function, like the probability of absence of crossings between the gas particle trajectory and the random field simulating the roughness.Statistical parameters of this model (verified by experiment) have better coincidence with the parameters of real micro-reliefs of applied in practice surfaces.E. g., poly-Gaussian model is successfully applied to simulate rough surfaces produced in different technology processes.
4. Solving both direct and inverse problems of surface roughness, the best way to reduce computing time is simulating the random field preliminary, before the main DSMC computation of rarefied gas flow.This structure of computational algorithm allows obtaining time economy for the same precision even in comparison with more simple models of roughness.The effect of aerodynamic shadowing on rough surface is studied for real surfaces (like Kapton and aluminium) and gases (like argon and helium).
5. The solution of inverse problem provides the values of σ 1 significantly higher (up to 1,5 times) than the profile diagram measurements.Thus, these calculations confirm the advantage of our method of accounting surface roughness on the base of the solution of inverse problem.

Figure 2 .
Figure 2. Continuum integral (9) as a function of the incidence angle θ (in deg.) and of the roughness parameter σ 1 .

Figure 4 .
Figure 4.The normal momentum exchange coefficients of reflected Ar and He atoms with various molecular energy of incident molecular beam and various roughness of Kapton or aluminium surface.
Poly-Gaussian model of surface roughness Gaussian homogeneous isotropic random fields have been studied as a model of the roughness in St.-Petersburg State University 2 2.