Question : Why in some languages, people pronounce दौड़ेगा daur̤egā (will run) as दौरेगा dauregā and why Indian English uses an ‘R’ in the spelling of Pakora or Chopra for a word like पकोड़ा pakor̤ā or चोपड़ा copr̤ā, where we can clearly realize that the pronunciation is not like an R ?
Answer : To trace the history of ड़ r̤ in Hindi or other modern Indo-Aryan languages, we have to go back to times when Old Indo Aryan (Sanskrit) intervocalic retroflex ṭ (ट्) & ṭh (ठ्) had systematically changed to r̤ (ड़) & ṛh (ढ़्), in Hindi if preceded by p (प्), ph (फ्), c (च्), ch (च्) & gh (घ्). Sanskrit पठनम् > पढ़ना (to read); पीठ (seat) > पीढ़ा (wooden stool), स्फोटक > फोड़ा (boil, abscess); चाटक > चिड़िया (bird); घट > घड़ा (clay pitcher). paṭhanam > par̤hnā; pīṭha > pīr̤hā, sphoṭaka > phor̤ā; cāṭaka > cir̤iyā; ghaṭa > ghar̤ā.
Diachronically Old Indo Aryan ṭ (ट) & ḍ (ड) changed to Middle Indo Aryan (Pali) ḍ (ड). In Modern Indo-Aryan, northwestern languages like Hindi, Punjabi, this ḍ (ड) is in complementary distribution (almost allophonic) with r̤ (ड़), in initial, geminate, and postnasal positions– e.g., Hindi डंडा ḍaṇḍā, whereas intervocalic, final, and before or after other consonants for r̤ (ड़) – e.g., Hindi घोड़ा ghor̤ā.
Interestingly this r̤ (ड़) is present phonemically in some languages (Western Hindi, Sindhi, Bengali etc.) that lack ḷ (ळ) and absent from some (Marathi, Gujarati) that possess ḷ (ळ).
In Odia which has ḷ (ळ) , voiced retroflex stops ḍ (ड) and ḍh (ढ) have allophones ṛ (ड़) and ṛh (ढ़). These flap allophones occur intervocalically and word finally.
I am not sure why r̤ (ड़) is orthographically preserved in language name ଓଡ଼ିଆ, even though I hear oḍia (ओडिया) in the audio link. Interestingly in the state emblem of Odisha, the word ଓଡ଼ିଶା or̥ishā is transliterated in Devanagari as ओडिशा oḍishā.
ṛ (ड़ ) is sub phonemic in Marathi, Gujarati, Eastern Hindi, Bhojpuri, Maithili, Kumauni, Kashmiri etc.) and is on the verge of merging with r (र) in Maithili. ṛ (ड़ ) is absent from Assamese, East & North Bengali dialects, and is remapped to r (र).
In Gujarati, it is absent in script but pronunciation is present in many words
कड़क (strong, stiff) > kar̤ak but in writing કડક (kaḍaka); कपड़ा (cloth) > kapr̤ā but in writing કપડા kapḍā and so on.
In Modern Standard Hindi Retroflex stop ḍ (ड) & Retroflex flap r̤ (ड़) is phonemic. That means they are recognized as separate phones & not as allophones. In earlier forms of Hindi, they were in complementary distribution i.e. ḍ (ड) was pronounced as r̤ (ड़) in intervocalic & final positions. This change is not so recent, it was present in times of Saint poet Tulsidas where he used /ḍ/ ड medially in निडर (niḍar = fearless). Now in Modern Standard Hindi, there are several minimal pairs which demarcate this difference –
पेड (english paid) X पेड़ peṛ (tree)
रोड (road) X रोड़ा roṛā (stone)
More examples in रेडियो, मोड, मोड़ reḍiyo, moḍ, mor̤
To add to it, this Retroflex stop ḍ (ड) & Retroflex flap r̤ (ड़) distinction is more evident in urban Hindi (मानक हिन्दी) rather than rural Hindi (ग्रामीण हिन्दी). It would be considered an error if somebody writes or pronounces पेड़ा or घोड़ा as पेडा or घोडा [pēṛā or ghōṛā as pēḍā or ghōḍā]; & रेडियो as रेड़ियो.
In Indian English, this Retroflex flap r̤ (ड़) is often mapped to an R as evident in words like Baroda (< बड़ौदा), Marwari (मारवाड़ी), Arora (अरोड़ा), Chopra (चोपड़ा), Burari (बुराड़ी) etc., although ṭ (ट) & ḍ (ड) are always written with T and D. My assumption is, the reason why r̤ (ड़) in Indian words is often mapped to an R in romanized spellings is because of the Hunterian transliteration system (1872) of romanization which was the first officially approved system of transliteration under British administration”. Under this system, r̤ (ड़) is romanized by an r /R. India and Pakistan continue to use the same romanization system for many place names – the Hunterian System – as they used before their independence.