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the fundamental problem of aerodynamics is to find the action force allocated along with the surface of a body in the flow ( such as the lift and drag force on airfoils , etc ) , thus we shall first study how the physical quantities distribute in the entire flow field . the relevant theory development peaked at the time shortly after joukowski and prandtl dedicated their effort . however navier - stoke equation and reynolds equation had become to their fellow researchers formidable mathematical obstacles ahead of them . in order to attain accurate data records in a flow field , we have to be aware of the complete figure of the field with great details , such as the regulatory development routine for the boundary layer , the laminar - turbulent transition , the development of the separation region , and the mechanism of turbulence energy dissipation in all sections . only through unveiling the mysteries of these principles , can we learn more about the pressure distribution as well as the appropriate circumstances for applying euler equations . when we study turbulence into depth , we shall refer to a few people of importance : g.i taylor , l. prandtl , von karman , kolmogorov , and r. kraichnann . it s worthy to note that , while potential flow theory and viscous flow theory can neither provide complete sustainable explanations for the physical principles behind the development of a flow field , the quantum flow field we discuss in this paper shall be recognized as a hypothesis . the studies we are doing today , apart from what our elder generation of researchers had aimed for , are to actualize the quantization of flow fields by incorporating a generalized de broglie relation . the diagram of the quantized flow field is as shown in _ figure 1_.        the topic regarding traditional flow field problems would focus on the physical quantity distributions of a uniform straight flow along with the surface of a rigid body in an arbitrary shape , and on the computation of the resulting force distributions acting on the particular object . similarly to the classical potential theory where we use a dipole instead of a sphere , or use the distribution function of a source / sink instead of an ordinary airfoil model , in the quantization theory of a flow field , we replace the uniform straight flow with a plane wave and the body in the field with a generalized potential function . but we ought to mention that here the plane wave , is in fact a generalized de broglie wave , i.e. a material wave ; is not an oscillation from any classical physics phenomena . once the geometrical figuration of an object in the flow field is steady to the observers in classical potential theory , there exists an exact potential function , which is the potential of velocity , and this potential function will be the solution of the entire flow field . in resemblance of the potential theory , the quantization method of a flow field thus seeks a particular wave function for some generalized schrodinger equation . mathematically speaking , the generalization of de broglie relation does nt encounter any difficulties in the effort doing logical advancement . the following equations hold up the fundamental structures in the quantization of a flow field :    1.generalized de broglie relation : @xmath0 @xmath1    2.generalized schrodinger equation : @xmath2\varphi = e_{\infty}\varphi\ ] ]    3.generalized velocity(in probability ) distributions : @xmath3\ ] ]    the above equations are the theoretical framework of a flow over a body under certain circumstances completely made up of material waves , as already shown in _ figure 1_. as indicated , @xmath4 and @xmath5 are respectively the quantity of the momentum and of the kinetic energy for the uniform straight flow in unit volume . and @xmath6 , where @xmath7 is the generalized planck constant , its according dimension is the fluid angular moment in the unit volume . @xmath8 is a generalized de broglie wavelength ; @xmath9 , @xmath10 , @xmath11 is the generalized potential function ; its dimension corresponds to the energy in unit volume ; similarly , the dimension for @xmath12 stands for the velocity . if we plug in the generalized de broglie relation ( 1 ) into the explicit expression for reynolds number : @xmath13 we can get the most significant dimensionless quantity in our wave features , i.e. the generalized re number : @xmath14    these are the _ similarity criteria _ for dynamics in a quantized environment , and in the above formula , both @xmath7 and @xmath15 are attached to various numerical values with respect to different fluid mediums ; in addition , they will have the same dimension . the existing and only admitted dimensionless parameter for all types of waves diffractions exhibits as @xmath16 . the ratio for @xmath17 settles the exact correspondence in coordinate parametrization between @xmath18 and @xmath19 . as shown in figure 2 , when we set the origin for polar coordinates to be the center of a sphere , the generalized potential function @xmath11 would have a simplified expression written in polar coordinates : @xmath20    and @xmath21 where @xmath22 is the radius of the spherical body . here @xmath23 is as required for the boundary condition . the formulas ( 5 ) and ( 6 ) are the _ no slipping condition _ and _ impenetrability condition _ on the surface of the body in a quantized flow field . noticing that the problem of the flow over a sphere , as we have discussed from above , has turned into a spherical symmetric generalized potential function acting on the plane wave presenting the uniform straight flow as a scattering process , and such a problem can have a relatively simple solution with partial wave method . at this moment the generalized schrodinger equation may have an asymptotic solution : @xmath24    where @xmath25 which is the so called scattering amplitude , and @xmath26 is the differential cross section . @xmath27 correspond to the partial waves @xmath28 etc . @xmath29 is the phase shift for the @xmath30 partial wave . and @xmath31 is the @xmath30 term for legendre polynomial . substituting ( 7 ) into ( 4 ) , we can obtain the asymptotic solution for the velocity distribution in probability , i.e the solution to the far - field wake behind the spherical body . because of our limitation on the computational skills and the enormous amount of work required for further analysis , here we only discuss the velocity distribution in probability parallel to the z - axis for some cross - section of a wake . its asymptotic solution has the form : @xmath32\ ] ] where @xmath33 , @xmath34 and @xmath35 are the real and imaginary parts for @xmath36 . this is an approximated one omitting some higher order terms when @xmath37 , _ figure 3 _ displays it in its typical prominence with 2 examples of @xmath38 ( as @xmath39 grows to @xmath40 ) and @xmath41 ( as @xmath39 grows to @xmath42 . )        in a graph carrying the main properties of the curve @xmath43 , it demonstrates which we shall consider as the basic form of what @xmath43 really is , the velocity distribution in terms of its probability natures , and thus we shall average @xmath43 in some way to get a result assimilating the curves being confirmed from wake attending experiments . but it s worth noting that even if we do nt take some form of average to the @xmath43 curve , there still always exists a _ deficit area _ on the graph for the velocity near z - axis ( when @xmath44 ) . when @xmath45 becomes big enough , we can see a special character called _ modulation _ in the `` tail '' ( i.e. perpendicular to the z - axis ) of the curve @xmath43 describing the velocity in a quantized wake . the _ `` modulation '' _ implies that when @xmath45 ( @xmath46 ) increases , wake width decreases , and it may also suggest there exists a multitude of _ wave packets _ in the flow field but outside of the wake . we call the indented parts between these wave packets _ wasp - waist shape _ ( when @xmath45 is small , we ca nt see a second wave packet on the @xmath43 curve so we may consider the _ transient point _ to be the _ wasp - waist shape _ as the first wave packet narrowing down until where its two horizontal `` boundary lines '' become relatively parallel . ) these mysterious figures of the velocity distribution in probability are first found in data analysis , which shall draw our special attention since they may be encrypted with unpredictable physical importance . we can at least propose that one among many of the quantum flow field properties is the complex interference between the bouquets of de broglie waves . in classical flow field theory , the transition mechanism turning from turbulence flow to laminar flow and the displacement of such a transition lack a creditable theoretical explanation , and therefore this becomes one of the key reasons why we are still having troubles determining the numerical values of the dynamical parameters ( especially for the lift and drag force parameters ) on bodies in a flow field . however , if we use the quantization method of the flow over a body , including formulas of computations in the entire flow field and even for the computations of the mechanism of emerging turbulence flow , we may lead ourselves to an overall new prospect for analysis of the flow field . in fact , @xmath47 i.e. the momentum distribution in probability can be extracted from the fourier transform of wave functions to the generalized schrodinger equation , which implies we may get the spectral function directly . this is already defined by de broglie relation @xmath48 . and it also constitutes the basics of the statistical description of turbulence , i.e. the concepts behind the energy spectral density function must come from the wave function in the momentum space , which is in resemblance with the turbulence spectral theory @xmath49 forwarded by g.i.taylor . in contrast to what g.i.taylor believed about the mechanism of energy - spectral analysis , we think that it s more reasonable and perceivable to consider the family of waves in momentum space to be the generalized de broglie waves , which is a physics cause in principle of the energy spectral density function . by the observation of some phenomenon in data analysis , we accommodate the u function into some form of its average and thus lead to the formula of the computation for drag under quantized environment : @xmath50 where @xmath51 is an undetermined constant , @xmath52 is the velocity distribution in probability and its average ( in quantum flow field ) , which is very close to the classical integral computational method for drag . @xmath53 stands for some necessary complementing term added to the drag formula upon the fact that the occurring pattern in the graph of velocity distribution in probability inside a wake is in correspondence to the energy of turbulence fluctuation . but mind that it differs from the energy dissipation term of the turbulence theory . given that we already have the simplified expression of a solution to the velocity in probability of flow over a sphere ( as shown in equation ( 9 ) ) , taking account of the existence of the _ wasp - waist shape _ between wave packets , we shall expect that @xmath53 having a mathematical form of : @xmath54 where @xmath55 are parameters involving contents from the _ wasp - waist shapes , _ which shall be chosen based on individual experiments using simulations .    in our data analysis by simulations , we emphasize some of the features which should be carefully studied in the future :    \1 ) when @xmath45 approaches 0 , the velocity distribution in probability would be back to the classical velocity profile as for @xmath56 , i.e. the width of the wake grows to infinity ( or similarly @xmath57 . ) this shows for @xmath58 or for @xmath56 the quantized flow field and the classical flow field would lead to the same solution for the velocity distribution . _ figure 4 _ shows the @xmath43 curves for @xmath59 with each curve rotated @xmath60 counter - clockwise .     counter - clockwise for various @xmath45 values _ _ ]    \2 ) if we pick out the first _ wasp - waist shapes _ from each @xmath43 graph and place them all together into one curve as _ figure 5 _ displays ( for 17 typical @xmath45 values : 0.000001 , 0.01 , 0.1 , 0.9 , 1.6 , 2.3 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 ) , we first see a drastic drop off between @xmath61 since we ca nt even see clear oscillating features especially when @xmath45 becomes smaller than 0.01 . so we may assert that the first _ wasp - waist shape _ appears very close to @xmath60 , in other words , close to _ infinity _ if we use cartesian coordinates . moreover the rate of change of the positions where the first wasp - waist shape figures appearing on @xmath43 curves getting closer to @xmath62 relatively accelerates for another time around the region @xmath63 . we hold that this is a primary reason for the existence of the drag crisis . if we leave the `` red curve '' on _ figure 5 _ only and adjust _ ka values _ to correct scales , i.e. @xmath64 , we will get a curve as depicted in the graph of _ figure 6_. but at this point our knowledge about the @xmath65 as well as where it would appear is minimal . so we should note that our reading to the locations of the first @xmath66 now is highly depending on visual premises , although our assumptions as follows would maintain some theoretical validity , this evaluation should only be considered as a hypothetical reference . notice the first wave packet does nt emerge distinctively until @xmath45 reaches @xmath67 , and the second wave packet appears when @xmath68 , so for @xmath69 we claim that the _ wasp - waist shape _ sits on the region separating the first wave packet from the `` tail '' , i.e. the part where the `` amplitude '' of the wave increases and the part where the amplitude stays relatively constant ; and for @xmath70 ( where the growth rate of the wave amplitude is nt quite distinguishable ) , we pick our point based on the same assumption as before but this reading may include some inconsistency needed to be clarified in future studies because of the limited information we know about the nature behind these wave packets and time required for computations in depth supporting the legitimate evidence .    _ ( note that in figure 5 and 6 we only compute the u values from @xmath62 to @xmath71 for @xmath45 is greater than @xmath72 whereas @xmath73 to @xmath74 for @xmath75 and @xmath73 to @xmath76 for @xmath77 due to the exponentially increasing number of points we need to select for computations as @xmath45 grows bigger . ) _    in quantum flow field , @xmath78 would attain a wide range of efficiency . this is related to the fact that the impact of s wave ( the component wave corresponding to the first term in the series embedded in the simplified solution of ( 8) ) on the cross - section of the scattering reduces to a much less significant extent , and causing by generalized ramsauer effect , reduces to 0 when @xmath45 reaches @xmath79 . the generalized ramsauer effect depicts that every component wave would encounter a _ transmission _ when @xmath45 equals certain values . then the cross - section would become smaller as a result of a transmission . thus this overall affects the value of the drag coefficient @xmath80 . notice that the _ wasp - waist shape _ on each @xmath43 curve is not unique , there exist multiple ones especially when @xmath45 becomes greater than 10 ( in our original calculation when @xmath45 grows up to 40 we saw as many as 4 of these shapes on each @xmath43 curve . ) this is a result emerging specifically in wave mechanics . the quantization method transforms the classical flow field into a capricious interfering wave family where the existence for modulation is a primary characteristic . from the figure we shall notice the undeniable importance of the first _ wasp waist shape _ to the calculation of drag coefficient ; thus it would be reasonable to predict the rest multiple wasp waist shapes on the @xmath43 curves when @xmath45 gets higher also devote into the formation of drag coefficients , i.e. in last integral formula presented above the @xmath81 term is an essential term directly related to the number or the properties of all the wasp waist shapes . but at the moment we could not provide precise information about them or about the numerical relations between the other multiple wasp waist shapes and the drag coefficient . in the future potential researches following this paper , we encourage people to specifically work on these features . boundary layer theory by prandtl is the most widely accepted theory for illustrating the actual process of flow over a body to our current researchers in the field of aerodynamics . yet we still lack some valid theoretical explanations for experimental data , i.e. when @xmath19 number grows greater than @xmath78 , why are both drag coefficient decreasing and its increasing gradual changes over the entire phase instead of sudden shifts ? prandtl s well known experiment was to place a thin metal ring to the windward side of a sphere and to make the separation point on the surface of that sphere move to the lower course ; it does nt provide coordinate - dependent demonstration especially on the time - space evolution for this separation process . on the contrary , suppose the transition from laminar flow to turbulence flow is a morphological abruptness , then there will be a great scale of topological changes , and @xmath78 is a relatively small region , further , @xmath80 will have a relatively minor change . however this does nt agree with experimental facts . not only @xmath80 changes over a wide range of efficiency , but its numerical value grows back slowly as @xmath19 becomes greater than @xmath82 . and the separation point on the surface of the sphere moves to the upper course instead . boundary layer theory has so far failed to provide valuable insights on this disagreement . the authors are sincerely grateful to the people for making this paper possible including family and friends . the late professors zhuang fenggan , and xu huafang , founders of china s studies in aerodynamics who had encouraged the authors for the development of this theory ; professor li chunxuan at beihang university who offered treasured consultation for advancing this paper ; professors zhang fuqing , liu ziqiang and yuan youxin for their detailed advice during the discussions about this paper ; mrs.tan liuning for her generous help financially and her first approach in some preliminary computations which turned out to be significant results .    pratyush sarkar , ethan yale jaffe , freid tong , michael yu , who all have been great friends and been utterly patient providing the author professional opinions regarding the paper and other mathematical topics ; yuqing tang , tom wang , and guanchu liu for their help initiating this paper ; li , andrew , tomas , zhifei , eric , yuan , david , christopher , adam , samer , jonathan , and all the other friends without whom this paper can not be possible .              formulas ( 4 ) to ( 8) @xcite have the almost identical presentations to the similar setting ups for the quantum scattering theory . the referenced books are the ones we used when consulting background knowledge from quantum mechanics @xcite @xcite , but we believe most of the earlier books studying quantum should all have these formulas and it s hard to track the true origin of them . the theoretical as well as the computational results presented in this paper do nt have relative fellow researchers work to be compared with , and the theory been developed behind this paper originated from the author yufei liu s book _ an introduction to quantum methods for flows around a body _ published in 2013 ( currently only formally available in mandarin . ) as almost all of the named theories or problems in the text aside from the new method introduced the first time are very well known and should not cause any confusion , we decided not to list any specific articles or books for references aside from the above ones .
in this paper we briefly introduce the quantum methods for computations of the drag coefficients for flows around a body , using the flows around a rigid sphere as an example , and we aim for comparing the wake under quantized environment and the classical computational strategies for finding the drag force . 
 this paper however does nt provide discussion on the pressure distribution over the surfaces of a spherical body and neither the analytical solution for the flows around an airfoil due to the greatly limited computational capacities and resources .